Please refer to the errata for this document, which may include some normative corrections.
In addition to the HTML version, this document is also available in these non-normative formats: diff marked HTML version, XHTML+MathML version, single page HTML5+MathML version, and PDF version.
See also translations.
Copyright © 1998-2014 W3C® (MIT, ERCIM, Keio, Beihang), All Rights Reserved. W3C liability, trademark and document use rules apply.
The presentation of this document has been augmented to use HTML5 rather than HTML4, and to include all MathML examples as inlined MathML markup that should be rendered by a MathML enabled browser. The MathML examples that have been added, and which should be displayed in your browser, are marked like this.
Note:
Most symbol characters in examples may be shown in one of three forms:
Entity reference (∑
),
Numeric character reference (∑
)
or as the character data directly (∑
).
By default, numeric references are used but you may alter the display by selecting an appropriate button below:
Note:
Your browser may be able to display Presentation MathML but not Content MathML.
You may convert the inline Content MathML to a Presentation MathML rendering using an XSLT transfomation.
This specification defines the Mathematical Markup Language, or MathML. MathML is a markup language for describing mathematical notation and capturing both its structure and content. The goal of MathML is to enable mathematics to be served, received, and processed on the World Wide Web, just as HTML has enabled this functionality for text.
This specification of the markup language MathML is intended primarily for a readership consisting of those who will be developing or implementing renderers or editors using it, or software that will communicate using MathML as a protocol for input or output. It is not a User's Guide but rather a reference document.
MathML can be used to encode both mathematical notation and mathematical content. About thirty-eight of the MathML tags describe abstract notational structures, while another about one hundred and seventy provide a way of unambiguously specifying the intended meaning of an expression. Additional chapters discuss how the MathML content and presentation elements interact, and how MathML renderers might be implemented and should interact with browsers. Finally, this document addresses the issue of special characters used for mathematics, their handling in MathML, their presence in Unicode, and their relation to fonts.
While MathML is human-readable, authors typically will use equation editors, conversion programs, and other specialized software tools to generate MathML. Several versions of such MathML tools exist, both freely available software and commercial products, and more are under development.
MathML was originally specified as an XML application and most of the examples in this specification assume that syntax. Other syntaxes are possible most notably [HTML5] specifies the syntax for MathML in HTML. Unless explictly noted, the examples in this specification are also valid HTML syntax.
This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.
This document was produced by the W3C Math Working Group as a Recommendation and is part of the W3C Math Activity. The goals of the W3C Math Working Group are discussed in the W3C Math WG Charter (revised July 2006). The authors of this document are the W3C Math Working Group members. A list of participants in the W3C Math Working Group is available.
This document has been reviewed by W3C Members, by software developers, and by other W3C groups and interested parties, and is endorsed by the Director as a W3C Recommendation. It is a stable document and may be used as reference material or cited from another document. W3C's role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability of the Web.
All reported errata to the first edition have been addressed in this addition, and a full change log appears in Appendix F Changes. The diff-marked version linked in the frontmatter highlights all changes between the first and second editions. In addition to incorporating errata, the main change in this addition is to recognise that MathML parsing is also specified in [HTML5] and where necessary to note where HTML and XML usage differ.
The Working Group maintains a comprehensive Test Suite. This is publicly available and developers are encouraged to submit their results for display. The Test Results are public. They show at least two interoperable implementations for each essential test. Further details may be found in the Implementation Report.
The MathML 2.0 (Second Edition) specification has been a W3C Recommendation since 2001. After its recommendation, a W3C Math Interest Group collected reports of experience with the deployment of MathML and identified issues with MathML that might be ameliorated. The rechartering of a Math Working Group did not signal any change in the overall design of MathML. The major additions in MathML 3 are support for bidirectional layout, better linebreaking and explicit positioning, elementary math notations, and a new strict content MathML vocabulary with well-defined semantics. The MathML 3 Specification has also been restructured.
This document was produced by a group operating under the 5 February 2004 W3C Patent Policy. W3C maintains a public list of any patent disclosures made in connection with the deliverables of the group; that page also includes instructions for disclosing a patent. An individual who has actual knowledge of a patent which the individual believes contains Essential Claim(s) must disclose the information in accordance with section 6 of the W3C Patent Policy.
Public discussion of MathML and issues of support through the W3C
for mathematics on the Web takes place on the public mailing list of the Math Working
Group (list archives).
To subscribe send an email to www-math-request@w3.org
with the word subscribe
in the subject line.
The basic chapter structure of this document is based on the earlier MathML 2.0 Recommendation [MathML2]. That MathML 2.0 itself was a revision of the earlier W3C Recommendation MathML 1.01 [MathML1]; MathML 3.0 is a revision of the W3C Recommendation MathML 2.0. It differs from it in that all previous chapters have been updated, some new elements and attributes added and some deprecated. Much has been moved to separate documents containing explanatory material, material on characters and entities and on the MathML DOM. The discussion of character entities has led to the document XML Entity Definitions for Characters [Entities], which is now a W3C Recommendation. The concern with use of CSS with MathML has led to the document A MathML for CSS Profile [MathMLforCSS], which was a W3C Recommendation accompanying MathML 3.0.
The biggest differences from MathML 2.0 (Second Edition) are in Chapters 4 and 5, although there have been smaller improvements throughout the specification. A more detailed description of changes from the previous Recommendation follows.
Much of the non-normative explication that formerly was found in Chapters 1 and 2, and many examples from elsewhere in the previous MathML specifications, were removed from the MathML3 specification and planned to be incorporated into a MathML Primer to be prepared as a separate document. It is expected this will help the use of this formal MathML3 specification as a reference document in implementations, and offer the new user better help in understanding MathML's deployment. The remaining content of Chapters 1 and 2 has been edited to reflect the changes elsewhere in the document, and in the rapidly evolving Web environment. Some of the text in them went back to early days of the Web and XML, and its explanations are now commonplace.
Chapter 3, on presentation-oriented markup, adds new material on linebreaking,
and on markup for elementary math notations used in many
countries (mstack
, mlongdiv
and other
associated elements). Other changes include revisions to
the mglyph
, mpadded
and maction
elements and significant unification and cleanup of attribute
values.
Earlier work, as recorded in the W3C Note Arabic
mathematical notation, has allowed clarification of the
relationship with bidirectional text and examples
with RTL text have been added.
Chapter 4, on content-oriented markup, contains major changes and additions. The meaning of the actual content remains as before in principle, but a lot of work has been done on expressing it better. A few new elements have been added.
Chapter 5 has been refined as its purpose has been further clarified to deal with the mixing of markup languages. This chapter deals, in particular, with interrelations of parts of the MathML specification, especially with presentation and content markup.
Chapter 6 is a new addition which deals with the issues of interaction of MathML with a host environment. This chapter deals with interrelations of the MathML specification with XML and HTML, but in the context of deployment on the Web. In particular there is a discussion of the interaction of CSS with MathML.
Chapter 7 replaces the previous Chapter 6, and has been rewritten and reorganized to reflect the new situation in regard to Unicode, and the changed W3C context with regard to named character entities. The new W3C specification XML Entity Definitions for Characters, which incorporates those used for mathematics has become a a W3C Recommendation, [Entities].
The Appendices, of which there are eight shown, have been reworked. Appendix A now contains the new RelaxNG schema for MathML3 as well as discussion of MathML3 DTD issues. Appendix B addresses media types associated with MathML and implicitly constitutes a request for the registration of three new ones, as is now standard for work from the W3C. Appendix C contains a new simplified and reconsidered Operator Dictionary. Appendices D, E, F, G and H contain similar non-normative material to that in the previous specification, now appropriately updated.
A fuller discussion of the document's evolution can be found in Appendix F Changes.
1 Introduction
1.1 Mathematics and its Notation
1.2 Origins and Goals
1.2.1 Design Goals of MathML
1.3 Overview
1.4 A First Example
2 MathML Fundamentals
2.1 MathML Syntax and Grammar
2.1.1 General Considerations
2.1.2 MathML and Namespaces
2.1.3 Children versus Arguments
2.1.4 MathML and Rendering
2.1.5 MathML Attribute Values
2.1.6 Attributes Shared by all MathML Elements
2.1.7 Collapsing Whitespace in Input
2.2 The Top-Level
<math>
Element
2.2.1 Attributes
2.2.2 Deprecated Attributes
2.3 Conformance
2.3.1 MathML Conformance
2.3.2 Handling of Errors
2.3.3 Attributes for unspecified data
3 Presentation Markup
3.1 Introduction
3.1.1 What Presentation Elements Represent
3.1.2 Terminology Used In This Chapter
3.1.3 Required Arguments
3.1.4 Elements with Special Behaviors
3.1.5 Directionality
3.1.6 Displaystyle and Scriptlevel
3.1.7 Linebreaking of Expressions
3.1.8 Warning about fine-tuning of presentation
3.1.9 Summary of Presentation Elements
3.1.10 Mathematics style attributes common to presentation elements
3.2 Token Elements
3.2.1
Token Element Content Characters, <mglyph/>
3.2.2 Mathematics style attributes common to token elements
3.2.3 Identifier <mi>
3.2.4 Number <mn>
3.2.5 Operator, Fence, Separator or Accent
<mo>
3.2.6 Text <mtext>
3.2.7 Space <mspace/>
3.2.8 String Literal <ms>
3.3 General Layout Schemata
3.3.1 Horizontally Group Sub-Expressions
<mrow>
3.3.2 Fractions <mfrac>
3.3.3 Radicals <msqrt>
, <mroot>
3.3.4 Style Change <mstyle>
3.3.5 Error Message <merror>
3.3.6 Adjust Space Around Content
<mpadded>
3.3.7 Making Sub-Expressions Invisible <mphantom>
3.3.8 Expression Inside Pair of Fences
<mfenced>
3.3.9 Enclose Expression Inside Notation
<menclose>
3.4 Script and Limit Schemata
3.4.1 Subscript <msub>
3.4.2 Superscript <msup>
3.4.3 Subscript-superscript Pair <msubsup>
3.4.4 Underscript <munder>
3.4.5 Overscript <mover>
3.4.6 Underscript-overscript Pair
<munderover>
3.4.7 Prescripts and Tensor Indices
<mmultiscripts>
,
<mprescripts/>
,
<none/>
3.5 Tabular Math
3.5.1 Table or Matrix
<mtable>
3.5.2 Row in Table or Matrix <mtr>
3.5.3 Labeled Row in Table or Matrix
<mlabeledtr>
3.5.4 Entry in Table or Matrix <mtd>
3.5.5 Alignment Markers
<maligngroup/>
, <malignmark/>
3.6 Elementary Math
3.6.1 Stacks of Characters <mstack>
3.6.2 Long Division <mlongdiv>
3.6.3 Group Rows with Similiar Positions <msgroup>
3.6.4 Rows in Elementary Math <msrow>
3.6.5 Carries, Borrows, and Crossouts <mscarries>
3.6.6 A Single Carry <mscarry>
3.6.7 Horizontal Line <msline/>
3.6.8 Elementary Math Examples
3.7 Enlivening Expressions
3.7.1 Bind Action to Sub-Expression
<maction>
3.8 Semantics and Presentation
4 Content Markup
4.1 Introduction
4.1.1 The Intent of Content Markup
4.1.2 The Structure and Scope of Content MathML Expressions
4.1.3 Strict Content MathML
4.1.4 Content Dictionaries
4.1.5 Content MathML Concepts
4.2 Content MathML Elements Encoding Expression Structure
4.2.1 Numbers <cn>
4.2.2 Content Identifiers <ci>
4.2.3 Content Symbols <csymbol>
4.2.4 String Literals <cs>
4.2.5 Function Application <apply>
4.2.6 Bindings and Bound Variables <bind>
and <bvar>
4.2.7 Structure Sharing <share>
4.2.8 Attribution via semantics
4.2.9 Error Markup <cerror>
4.2.10 Encoded Bytes <cbytes>
4.3 Content MathML for Specific Structures
4.3.1 Container Markup
4.3.2 Bindings with <apply>
4.3.3 Qualifiers
4.3.4 Operator Classes
4.3.5 Non-strict Attributes
4.4 Content MathML for Specific Operators and Constants
4.4.1 Functions and Inverses
4.4.2 Arithmetic, Algebra and Logic
4.4.3 Relations
4.4.4 Calculus and Vector Calculus
4.4.5 Theory of Sets
4.4.6 Sequences and Series
4.4.7 Elementary classical functions
4.4.8 Statistics
4.4.9 Linear Algebra
4.4.10 Constant and Symbol Elements
4.5 Deprecated Content Elements
4.5.1 Declare <declare>
4.5.2 Relation <reln>
4.5.3 Relation <fn>
4.6 The Strict Content MathML Transformation
5 Mixing Markup Languages for Mathematical Expressions
5.1 Annotation Framework
5.1.1 Annotation elements
5.1.2 Annotation keys
5.1.3 Alternate representations
5.1.4 Content equivalents
5.1.5 Annotation references
5.2 Elements for Semantic Annotations
5.2.1 The <semantics>
element
5.2.2 The <annotation>
element
5.2.3 The <annotation-xml>
element
5.3 Combining Presentation and Content Markup
5.3.1 Presentation Markup in Content Markup
5.3.2 Content Markup in Presentation Markup
5.4 Parallel Markup
5.4.1 Top-level Parallel Markup
5.4.2 Parallel Markup via Cross-References
6 Interactions with the Host Environment
6.1 Introduction
6.2 Invoking MathML Processors
6.2.1 Recognizing MathML in XML
6.2.2 Recognizing MathML in HTML
6.2.3 Resource Types for MathML Documents
6.2.4 Names of MathML Encodings
6.3 Transferring MathML
6.3.1 Basic Transfer Flavor Names and Contents
6.3.2 Recommended Behaviors when Transferring
6.3.3 Discussion
6.3.4 Examples
6.4 Combining MathML and Other Formats
6.4.1 Mixing MathML and XHTML
6.4.2 Mixing MathML and non-XML contexts
6.4.3 Mixing MathML and HTML
6.4.4 Linking
6.4.5 MathML and Graphical Markup
6.5 Using CSS with MathML
6.5.1 Order of processing attributes versus style sheets
7 Characters, Entities and Fonts
7.1 Introduction
7.2 Unicode Character Data
7.3 Entity Declarations
7.4 Special Characters Not in Unicode
7.5 Mathematical Alphanumeric Symbols
7.6 Non-Marking Characters
7.7 Anomalous Mathematical Characters
7.7.1 Keyboard Characters
7.7.2 Pseudo-scripts
7.7.3 Combining Characters
A Parsing MathML
A.1 Use of MathML as Well-Formed
XML
A.2 Using the RelaxNG Schema for MathML3
A.2.1 Full MathML
A.2.2 Elements Common to Presentation and Content MathML
A.2.3 The Grammar for Presentation MathML
A.2.4 The Grammar for Strict Content MathML3
A.2.5 The Grammar for Content MathML
A.2.6 MathML as a module in a RelaxNG Schema
A.3 Using the MathML DTD
A.3.1 Document Validation Issues
A.3.2 Attribute values in the MathML DTD
A.3.3 DOCTYPE declaration for MathML
A.4 Using the MathML XML Schema
A.4.1 Associating the MathML schema with MathML fragments
A.5 Parsing MathML in XHTML
A.6 Parsing MathML in HTML
B Media Types Registrations
B.1 Selection of Media Types for
MathML Instances
B.2 Media type for Generic MathML
B.3 Media type for Presentation MathML
B.4 Media type for Content MathML
C Operator Dictionary (Non-Normative)
C.1 Indexing of the operator dictionary
C.2 Format of operator dictionary entries
C.3 Notes on lspace and
rspace attributes
C.4 Operator dictionary entries
D Glossary (Non-Normative)
E Working Group Membership and Acknowledgments (Non-Normative)
E.1 The Math Working Group Membership
E.2 Acknowledgments
F Changes (Non-Normative)
F.1 Changes between MathML 3.0 First Edition and Second Edition
F.2 Changes between MathML 2.0 Second Edition and MathML 3.0
G Normative References
H References (Non-Normative)
I Index (Non-Normative)
I.1 MathML Elements
I.2 MathML Attributes
A distinguishing feature of mathematics is the use of a complex and highly evolved system of two-dimensional symbolic notation. As J. R. Pierce writes in his book on communication theory, mathematics and its notation should not be viewed as one and the same thing [Pierce1961]. Mathematical ideas can exist independently of the notation that represents them. However, the relation between meaning and notation is subtle, and part of the power of mathematics to describe and analyze derives from its ability to represent and manipulate ideas in symbolic form. The challenge before a Mathematical Markup Language (MathML) in enabling mathematics on the World Wide Web is to capture both notation and content (that is, its meaning) in such a way that documents can utilize the highly evolved notation of written and printed mathematics as well as the new potential for interconnectivity in electronic media.
Mathematical notation evolves constantly as people continue to innovate in ways of approaching and expressing ideas. Even the common notation of arithmetic has gone through an amazing variety of styles, including many defunct ones advocated by leading mathematical figures of their day [Cajori1928]. Modern mathematical notation is the product of centuries of refinement, and the notational conventions for high-quality typesetting are quite complicated and subtle. For example, variables and letters which stand for numbers are usually typeset today in a special mathematical italic font subtly distinct from the usual text italic; this seems to have been introduced in Europe in the late sixteenth century. Spacing around symbols for operations such as +, -, × and / is slightly different from that of text, to reflect conventions about operator precedence that have evolved over centuries. Entire books have been devoted to the conventions of mathematical typesetting, from the alignment of superscripts and subscripts, to rules for choosing parenthesis sizes, and on to specialized notational practices for subfields of mathematics. The manuals describing the nuances of present-day computer typesetting and composition systems can run to hundreds of pages.
Notational conventions in mathematics, and in printed text in general,
guide the eye and make printed expressions much easier to read and
understand. Though we usually take them for granted, we, as modern readers,
rely on
numerous conventions such as paragraphs, capital letters, font
families and cases, and even the device of decimal-like numbering of
sections such as is used in this document.
Such notational conventions are perhaps even more important
for electronic media, where one must contend with the difficulties of
on-screen reading. Appropriate standards coupled with computers
enable a broadening of access to mathematics beyond the world of
print. The markup methods for mathematics in use
just before the Web rose to prominence importantly included TEX
(also written TeX
) [Knuth1986]
and approaches based on SGML ([AAP-math], [Poppelier1992] and [ISO-12083]).
It is remarkable how widespread the current conventions of mathematical notation have become. The general two-dimensional layout, and most of the same symbols, are used in all modern mathematical communications, whether the participants are, say, European, writing left-to-right, or Middle-Eastern, writing right-to-left. Of course, conventions for the symbols used, particularly those naming functions and variables, may tend to favor a local language and script. The largest variation from the most common is a form used in some Arabic-speaking communities which lays out the entire mathematical notation from right-to-left, roughly in mirror image of the European tradition.
However, there is more to putting mathematics on the Web than merely finding ways of displaying traditional mathematical notation in a Web browser. The Web represents a fundamental change in the underlying metaphor for knowledge storage, a change in which interconnection plays a central role. It has become important to find ways of communicating mathematics which facilitate automatic processing, searching and indexing, and reuse in other mathematical applications and contexts. With this advance in communication technology, there is an opportunity to expand our ability to represent, encode, and ultimately to communicate our mathematical insights and understanding with each other. We believe that MathML as specified below is an important step in developing mathematics on the Web.
MathML has been designed from the beginning with the following ultimate goals in mind.
MathML should ideally:
Encode mathematical material suitable for all educational and scientific communication.
Encode both mathematical notation and mathematical meaning.
Facilitate conversion to and from other mathematical formats, both presentational and semantic. Output formats should include:
graphical displays
speech synthesizers
input for computer algebra systems
other mathematics typesetting languages, such as TEX
plain text displays, e.g. VT100 emulators
international print media, including braille
It is recognized that conversion to and from other notational systems or media may entail loss of information in the process.
Allow the passing of information intended for specific renderers and applications.
Support efficient browsing of lengthy expressions.
Provide for extensibility.
Be well suited to templates and other common techniques for editing formulas.
Be legible to humans, and simple for software to generate and process.
No matter how successfully MathML achieves its goals as a markup language, it is clear that MathML is useful only if it is implemented well. The W3C Math Working Group has identified a short list of additional implementation goals. These goals attempt to describe concisely the minimal functionality MathML rendering and processing software should try to provide.
MathML expressions in HTML (and XHTML) pages should render properly in popular Web browsers, in accordance with reader and author viewing preferences, and at the highest quality possible given the capabilities of the platform.
HTML (and XHTML) documents containing MathML expressions should print properly and at high-quality printer resolutions.
MathML expressions in Web pages should be able to react to user gestures, such those as with a mouse, and to coordinate communication with other applications through the browser.
Mathematical expression editors and converters should be developed to facilitate the creation of Web pages containing MathML expressions.
The extent to which these goals are ultimately met depends on the cooperation and support of browser vendors and other developers. The W3C Math Working Group has continued to work with other working groups of the W3C, and outside the W3C, to ensure that the needs of the scientific community will be met. MathML 2 and its implementations showed considerable progress in this area over the situation that obtained at the time of the MathML 1.0 Recommendation (April 1998) [MathML1]. MathML3 and the developing Web are expected to allow much more.
MathML is a markup language for describing mathematics. It is usually expressed in XML syntax, although HTML and other syntaxes are possible. A special aspect of MathML is that there are two main strains of markup: Presentation markup, discussed in Chapter 3 Presentation Markup, is used to display mathematical expressions; and Content markup, discussed in Chapter 4 Content Markup, is used to convey mathematical meaning. Content markup is specified in particular detail. This specification makes use of an XML format called Content Dictionaries This format has been developed by the OpenMath Society, [OpenMath2004] with the dictionaries being used by this specification involving joint development by the OpenMath Society and the W3C Math Working Group.
Fundamentals common to both strains of markup are covered in Chapter 2 MathML Fundamentals, while the means for combining these strains, as well as external markup, into single MathML objects are discussed in Chapter 5 Mixing Markup Languages for Mathematical Expressions. How MathML interacts with applications is covered in Chapter 6 Interactions with the Host Environment. Finally, a discussion of special symbols, and issues regarding characters, entities and fonts, is given in Chapter 7 Characters, Entities and Fonts.
The quadratic formula provides a simple but instructive illustration of MathML markup.
MathML offers two flavors of markup of this formula. The first is the style which emphasizes the actual presentation of a formula, the two-dimensional layout in which the symbols are arranged. An example of this type is given just below. The second flavor emphasizes the mathematical content and an example of it follows the first one.
<mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>-</mo> <mi>b</mi> </mrow> <mo>±<!--PLUS-MINUS SIGN--></mo> <msqrt> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>-</mo> <mrow> <mn>4</mn> <mo>⁢<!--INVISIBLE TIMES--></mo> <mi>a</mi> <mo>⁢<!--INVISIBLE TIMES--></mo> <mi>c</mi> </mrow> </mrow> </msqrt> </mrow> <mrow> <mn>2</mn> <mo>⁢<!--INVISIBLE TIMES--></mo> <mi>a</mi> </mrow> </mfrac> </mrow> |
Consider the superscript 2 in this formula. It represents the squaring operation here, but the meaning of a superscript in other situations depends on the context. A letter with a superscript can be used to signify a particular component of a vector, or maybe the superscript just labels a different type of some structure. Similarly two letters written one just after the other could signify two variables multiplied together, as they do in the quadratic formula, or they could be two letters making up the name of a single variable. What is called Content Markup in MathML allows closer specification of the mathematical meaning of many common formulas. The quadratic formula given in this style of markup is as follows.
<apply> <eq/> <ci>x</ci> <apply> <divide/> <apply> <plus/> <apply> <minus/> <ci>b</ci> </apply> <apply> <root/> <apply> <minus/> <apply> <power/> <ci>b</ci> <cn>2</cn> </apply> <apply> <times/> <cn>4</cn> <ci>a</ci> <ci>c</ci> </apply> </apply> </apply> </apply> <apply> <times/> <cn>2</cn> <ci>a</ci> </apply> </apply> </apply> |
The basic ‘syntax’ of MathML is defined using XML syntax, but other syntaxes that can encode labeled trees are possible. Notably the HTML parser may also be used with MathML. Upon this, we layer a ‘grammar’, being the rules for allowed elements, the order in which they can appear, and how they may be contained within each other, as well as additional syntactic rules for the values of attributes. These rules are defined by this specification, and formalized by a RelaxNG schema [RELAX-NG]. The RelaxNG Schema is normative, but a DTD (Document Type Definition) and an XML Schema [XMLSchemas] are provided for continuity (they were normative for MathML2). See Appendix A Parsing MathML.
MathML's character set consists of legal characters as specified by Unicode [Unicode], further restricted by the characters not allowed in XML. The use of Unicode characters for mathematics is discussed in Chapter 7 Characters, Entities and Fonts.
The following sections discuss the general aspects of the MathML grammar as well as describe the syntaxes used for attribute values.
An XML namespace [Namespaces] is a collection of names identified by a URI. The URI for the MathML namespace is:
http://www.w3.org/1998/Math/MathML
To declare a namespace when using the XML serialisation of MathML,
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 child elements. For example:
<math xmlns="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.
When embedding MathML within XHTML, one might use:
<body xmlns:m="http://www.w3.org/1998/Math/MathML"> ... <m:math><m:mrow>...</m:mrow></m:math> ... </body>
HTML does not support namespace extensibility in the same way, the HTML parser
has in-built knowledge of the HTML, SVG and MathML namespaces. xmlns
attributes are
just treated as normal attributes. Thus when using the HTML serialisation of MathML,
prefixed element names must not be used. xmlns
="http://www.w3.org/1998/Math/MathML"
may be used on the math element
, it will be ignored by the HTML parser,
which always places math
elements and its descendents in the MathML namespace
(other than special rules described in Appendix A Parsing MathMLfor invalid input, and for annotation-xml
.
If a MathML expression is likely to be in contexts where it may be parsed by an XML parser or an HTML parser, it SHOULD
use the following form to ensure maximum compatibility:
<math xmlns="http://www.w3.org/1998/Math/MathML"> ... </math>
Most MathML elements act as ‘containers’; such an element's
children are not distinguished from each other except as individual members of the
list of children. Commonly there is no limit imposed on the number of children
an element may have. This is the case for most presentation
elements and some content elements such as set
.
But many
MathML elements require a specific number of children, or
attach a particular meaning to children in certain positions.
Such elements are best considered to represent constructors of mathematical
objects, and hence thought of as functions of their children. Therefore
children of such a MathML element
will often be referred to as its arguments instead of merely as children.
Examples of this can be found, say, in Section 3.1.3 Required Arguments.
There are presentation elements that conceptually accept only
a single argument, but which for convenience have been written to accept any number of children;
then we infer an mrow
containing those children which acts as
the argument to the element in question; see Section 3.1.3.1 Inferred <mrow>
s.
In the detailed discussions of element syntax given with each element throughout the MathML specification, the correspondence of children with arguments, the number of arguments required and their order, as well as other constraints on the content, are specified. This information is also tabulated for the presentation elements in Section 3.1.3 Required Arguments.
MathML presentation elements only recommend (i.e., do not require) specific ways of rendering; this is in order to allow for medium-dependent rendering and for individual preferences of style.
Nevertheless, some parts of this specification describe these recommended visual rendering rules in detail; in those descriptions it is often assumed that the model of rendering used supports the concepts of a well-defined 'current rendering environment' which, in particular, specifies a 'current font', a 'current display' (for pixel size) and a 'current baseline'. The 'current font' provides certain metric properties and an encoding of glyphs.
MathML elements take attributes with values that further specialize
the meaning or effect of the element. Attribute names are shown in a
monospaced
font throughout this document. The meanings of attributes and their
allowed values are described within the specification of each element.
The syntax notation explained in this section is used in specifying allowed values.
Except when explicitly forbidden by the specification for an attribute, MathML attribute values may contain any legal characters specified by the XML recommendation. See Chapter 7 Characters, Entities and Fonts for further clarification.
To describe the MathML-specific syntax of attribute values, the following conventions and notations are used for most attributes in the present document. We use below the notation beginning with U+ that is recommended by Unicode for referring to Unicode characters [see [Unicode], page xxviii].
Notation | What it matches |
---|---|
decimal-digit | a decimal digit from the range U+0030 to U+0039 |
hexadecimal-digit | a hexadecimal (base 16) digit from the ranges U+0030 to U+0039, U+0041 to U+0046 and U+0061 to U+0066 |
unsigned-integer | a string of decimal-digits, representing a non-negative integer |
positive-integer | a string of decimal-digits, but not consisting solely of "0"s (U+0030), representing a positive integer |
integer | an optional "-" (U+002D), followed by a string of decimal digits, and representing an integer |
unsigned-number | a string of decimal digits with up to one decimal point (U+002E), representing a non-negative terminating decimal number (a type of rational number) |
number | an optional prefix of "-" (U+002D), followed by an unsigned number, representing a terminating decimal number (a type of rational number) |
character | a single non-whitespace character |
string | an arbitrary, nonempty and finite, string of characters |
length | a length, as explained below, Section 2.1.5.2 Length Valued Attributes |
unit | a unit, typically used as part of a length, as explained below, Section 2.1.5.2 Length Valued Attributes |
namedlength | a named length, as explained below, Section 2.1.5.2 Length Valued Attributes |
color | a color, as explained below, Section 2.1.5.3 Color Valued Attributes |
id | an identifier, unique within the document; must satisfy the NAME syntax of the XML recommendation [XML] |
idref | an identifier referring to another element within the document; must satisfy the NAME syntax of the XML recommendation [XML] |
URI | a Uniform Resource Identifier [RFC3986]. Note that the attribute value is typed in the schema as anyURI which allows any sequence of XML characters. Systems needing to use this string as a URI must encode the bytes of the UTF-8 encoding of any characters not allowed in URI using %HH encoding where HH are the byte value in hexadecimal. This ensures that such an attribute value may be interpreted as an IRI, or more generally a LEIRI, see [IRI]. |
italicized word | values as explained in the text for each attribute; see Section 2.1.5.4 Default values of attributes |
"literal" | quoted symbol, literally present in the attribute value (e.g. "+" or '+') |
The ‘types’ described above, except for string, may be combined into composite patterns using the following operators. The whole attribute value must be delimited by single (') or double (") quotation marks in the marked up document. Note that double quotation marks are often used in this specification to mark up literal expressions; an example is the "-" in line 5 of the table above.
In the table below a form f means an instance of a type described in the table above. The combining operators are shown in order of precedence from highest to lowest:
Notation | What it matches |
---|---|
( f ) | same as f |
f? |
an optional instance of f |
f * |
zero or more instances of f, with separating whitespace characters |
f + | one or more instances of f, with separating whitespace characters |
f1 f2 ... fn | one instance of each form fi, in sequence, with no separating whitespace |
f1, f2, ..., fn | one instance of each form fi, in sequence, with separating whitespace characters (but no commas) |
f1 | f2 | ... | fn | any one of the specified forms fi |
The notation we have chosen here is in the style of the syntactical notation of the RelaxNG used for MathML's basic schema, Appendix A Parsing MathML.
Since some applications are inconsistent about normalization of whitespace, for maximum interoperability it is advisable to use only a single whitespace character for separating parts of a value. Moreover, leading and trailing whitespace in attribute values should be avoided.
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.
If a numerical value within an attribute value syntax description
is declared to allow a minus sign ('-'), e.g., number
or
integer
, 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).
Most presentation elements have attributes that accept values representing lengths to be used for size, spacing or similar properties. The syntax of a length is specified as
Type | Syntax |
---|---|
length | number | number unit | namedspace |
There should be no space between the number and the unit of a length.
The possible units and namedspaces, along with their interpretations, are
shown below. Note that although the units and their meanings are taken from
CSS, the syntax of lengths is not identical. A few MathML elements
have length attributes that accept additional keywords; these are termed pseudo-units
and specified
in the description of those particular elements; see, for instance, Section 3.3.6 Adjust Space Around Content
<mpadded>
.
A trailing "%" represents a percent of a reference value; unless otherwise stated, the reference value is the default value. The default value, or how it is obtained, is listed in the table of attributes for each element along with the reference value when it differs from the default. (See also Section 2.1.5.4 Default values of attributes.) A number without a unit is intepreted as a multiple of the reference value. This form is primarily for backward compatibility and should be avoided, prefering explicit units for clarity.
In some cases, the range of acceptable values for a particular attribute may be restricted; implementations are free to round up or down to the closest allowable value.
The possible units in MathML are:
Unit | Description |
---|---|
em |
an em (font-relative unit traditionally used for horizontal lengths) |
ex |
an ex (font-relative unit traditionally used for vertical lengths) |
px |
pixels, or size of a pixel in the current display |
in |
inches (1 inch = 2.54 centimeters) |
cm |
centimeters |
mm |
millimeters |
pt |
points (1 point = 1/72 inch) |
pc |
picas (1 pica = 12 points) |
% |
percentage of the default value |
Some additional aspects of units are discussed further below, in Section 2.1.5.2.1 Additional notes about units.
The following constants, namedspaces, may also be used where a length is needed; they are typically used for spacing or padding between tokens. Recommended default values for these constants are shown; the actual spacing used is implementation specific.
namedspace | Recommended default |
---|---|
"veryverythinmathspace" | 1/18em |
"verythinmathspace" | 2/18em |
"thinmathspace" | 3/18em |
"mediummathspace" | 4/18em |
"thickmathspace" | 5/18em |
"verythickmathspace" | 6/18em |
"veryverythickmathspace" | 7/18em |
"negativeveryverythinmathspace" | -1/18em |
"negativeverythinmathspace" | -2/18em |
"negativethinmathspace" | -3/18em |
"negativemediummathspace" | -4/18em |
"negativethickmathspace" | -5/18em |
"negativeverythickmathspace" | -6/18em |
"negativeveryverythickmathspace" | -7/18em |
Lengths are only used in MathML for presentation, and presentation
will ultimately involve rendering in or on some medium. For visual media,
the display context is assumed to have certain properties available to
the rendering agent. A px
corresponds to a pixel on the display, to
the extent that is meaningful. The resolution of the display device
will affect the correspondence of pixels to the units
in
, cm
, mm
, pt
and pc
.
Moreover, the display context will also provide a default for the font size;
the parameters of this font determine the initial values used to interpret
the units em
and ex
, and thus indirectly the sizes
of namedspaces. Since these units track the display context, and in particular,
the user's preferences for display, the relative units em
and ex
are generally to be preferred over absolute units such as px
or cm
.
Two additional aspects of relative units must be clarified, however.
First, some elements such as Section 3.4 Script and Limit Schemata or mfrac
,
implicitly switch to smaller font sizes for some of their arguments.
Similarly, mstyle
can be used to explicitly change
the current font size. In such cases, the effective values of
an em
or ex
inside those contexts will be
different than outside. The second point is that the effective value
of an em
or ex
used for an attribute value
can be affected by changes to the current font size.
Thus, attributes that affect the current font size,
such as mathsize
and scriptlevel
, must be processed before
evaluating other length valued attributes.
If, and how, lengths might affect non-visual media is implementation specific.
The color, or background color, of presentation elements may be specified as a color using the following syntax:
Type | Syntax |
---|---|
color | #RGB | #RRGGBB | html-color-name |
A color is specified either by "#" followed
by hexadecimal values for the red, green, and blue components,
with no intervening whitespace, or by an html-color-name.
The color components can be either 1-digit or 2-digit, but
must all have the same number of digits; the component
ranges from 0 (component not present) to FF
(component fully present).
Note that, for example, by the digit-doubling rule specified under Colors in
[CSS21]
#123
is a short form for #112233
.
Color values can also be specified as an html-color-name, one of the color-name keywords defined in [HTML4] ("aqua", "black", "blue", "fuchsia", "gray", "green", "lime", "maroon", "navy", "olive", "purple", "red", "silver", "teal", "white", and "yellow"). Note that the color name keywords are not case-sensitive, unlike most keywords in MathML attribute values, for compatibility with CSS and HTML.
When a color is applied to an element,
it is the color in which the content of tokens is rendered.
Additionally, when inherited from a surrounding element or from the environment in which the complete MathML expression is
embedded, it controls the color of
all other drawing due to MathML elements, including the lines
or radical signs that can be drawn in rendering mfrac
, mtable
, or
msqrt
.
When used to specify a background color, the keyword "transparent"
is also allowed.
The recommended MathML visual rendering rules do not define the
precise extent of the region whose background is affected by using the
background
attribute on an element,
except that, when the element's content does not have
negative dimensions and its drawing region is not overlapped by other
drawing due to surrounding negative spacing, this region should lie
behind all the drawing done to render the content of the
element, but should not lie behind any of the
drawing done to render surrounding expressions. The effect of overlap
of drawing regions caused by negative spacing on the extent of the
region affected by the background
attribute is not
defined by these rules.
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, but when italicized are descriptions of how default values can be computed.
Default values described as inherited are taken from the
rendering environment, as described in Section 3.3.4 Style Change <mstyle>
,
or in some cases (which are described individually) taken 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.)
Default values described as automatic 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 context in which the element is rendered.
Other italicized descriptions of default values which appear in the tables of attributes are explained individually for each attribute.
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 around attribute values in examples in the text, so that the pieces of code shown are correct.
Note that, in general, there is no mechanism in MathML to simulate the
effect of not specifying attributes which are inherited or
automatic. Giving the words "inherited" or
"automatic" explicitly will not work, and is not generally
allowed. Furthermore, the mstyle
element (Section 3.3.4 Style Change <mstyle>
)
can even be used to change the default values of presentation attributes
for its children.
Note also that these defaults describe the behavior of MathML applications when an attribute is not supplied; they do not indicate a value that will be filled in by an XML parser, as is sometimes mandated by DTD-based specifications.
In general, there are a number of
properties of MathML rendering that may be thought of as overall
properties of a document, or at least of sections of a large
document. Examples might be mathsize
(the math font
size: see Section 3.2.2 Mathematics style attributes common to token elements), or the
behavior in setting limits on operators such as integrals or sums
(e.g., movablelimits
or displaystyle
), or
upon breaking formulas over lines (e.g.
linebreakstyle
); for such attributes see several
elements in Section 3.2 Token Elements.
These may be thought to be inherited from some such
containing scope. Just above we have mentioned the setting of default
values of MathML attributes as inherited or
automatic; there is a third source of global default values
for behavior in rendering MathML, a MathML operator dictionary. A
default example is provided in Appendix C Operator Dictionary.
This is also discussed in Section 3.2.5.7.1 The operator dictionary and examples are given in
Section 3.2.5.2.1 Dictionary-based attributes.
In addition to the attributes described specifically for each element,
the attributes in the following table are allowed on every MathML element.
Also allowed are attributes from the xml namespace, such as xml:lang
,
and attributes from namespaces other than MathML,
which are ignored by default.
Name | values | default |
---|---|---|
id | id | none |
Establishes a unique identifier associated with the element
to support linking, cross-references and parallel markup.
See xref and Section 5.4 Parallel Markup.
|
||
xref | idref | none |
References another element within the document.
See id and Section 5.4 Parallel Markup.
|
||
class | string | none |
Associates the element with a set of style classes for use with [XSLT] and [CSS21]. Typically this would be a space separated sequence of words, but this is not specified by MathML. See Section 6.5 Using CSS with MathML for discussion of the interaction of MathML and CSS. | ||
style | string | none |
Associates style information with the element for use with [XSLT] and [CSS21]. This typically would be an inline CSS style, but this is not specified by MathML. See Section 6.5 Using CSS with MathML for discussion of the interaction of MathML and CSS. | ||
href | URI | none |
Can be used to establish the element as a hyperlink to the specfied URI. |
Note that MathML 2 had no direct support for linking, and instead
followed the W3C Recommendation "XML Linking Language"
[XLink] in defining links using the
xlink:href
attribute. This has changed, and MathML 3 now
uses an href
attribute. However, particular compound
document formats may specify the use of XML linking with MathML
elements, so user agents that support XML linking should continue to
support the use of the xlink:href
attribute with MathML 3
as well.
See also Section 3.2.2 Mathematics style attributes common to token elements for a list of MathML attributes which can be used on most presentation token elements.
The attribute other
,
is deprecated
(Section 2.3.3 Attributes for unspecified data) in favor of the use of
attributes from other namespaces.
Name | values | default |
---|---|---|
other | string | none |
DEPRECATED but in MathML 1.0. |
In MathML, as in XML, "whitespace" means simple spaces, tabs, newlines, or carriage returns, i.e., characters with hexadecimal Unicode codes U+0020, U+0009, U+000A, or U+000D, respectively; see also the discussion of whitespace in Section 2.3 of [XML].
MathML ignores whitespace occurring outside token elements.
Non-whitespace characters are not allowed there. Whitespace occurring
within the content of token elements , except for <cs>
, is normalized as follows. All whitespace at the beginning and end of the content is
removed, and whitespace internal to content of the element is
collapsed canonically, i.e., each sequence of 1 or more
whitespace characters is replaced with one space character (U+0020, sometimes
called a blank character).
For example, <mo> ( </mo>
is equivalent to
<mo>(</mo>
, and
<mtext> Theorem 1: </mtext> |
is equivalent to
<mtext>Theorem 1:</mtext>
or
<mtext>Theorem 1:</mtext>
.
Authors wishing to encode white space 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
(U+00A0)
or other non-marking characters that are not trimmed.
For example, compare the above use of an mtext
element
with
<mtext>  <!--NO-BREAK SPACE-->Theorem  <!--NO-BREAK SPACE-->1: </mtext> |
When the first example is rendered, there is nothing before "Theorem", one Unicode space character between "Theorem" and "1:", and nothing after "1:". In the second example, a single space character is to be rendered before "Theorem"; two spaces, one a Unicode space character and one a Unicode no-break space character, are to be rendered before "1:"; and there is nothing after the "1:".
Note that the value of the xml:space
attribute is not relevant
in this situation since XML processors pass whitespace in tokens to a
MathML processor; it is the requirements of MathML processing which specify that
whitespace is trimmed and collapsed.
For whitespace occurring outside the content of the token elements
mi
, mn
, mo
, ms
, mtext
,
ci
, cn
, cs
, csymbol
and annotation
,
an mspace
element should be used, as opposed to an mtext
element containing
only whitespace entities.
<math>
Element
MathML 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; in other words,
every valid MathML expression is wrapped in outer
<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. These considerations also apply when sub-expressions are
passed between applications, such as for cut-and-paste operations;
See Section 6.3 Transferring MathML.
The math
element can contain an arbitrary number
of child elements. They render by default as if they
were contained in an mrow
element.
The math
element accepts any of the attributes that can be set on
Section 3.3.4 Style Change <mstyle>
, including the common attributes
specified in Section 2.1.6 Attributes Shared by all MathML Elements.
In particular, it accepts the dir
attribute for
setting the overall directionality; the math
element is usually
the most useful place to specify the directionality
(See Section 3.1.5 Directionality for further discussion).
Note that the dir
attribute defaults to "ltr"
on the math
element (but inherits on all other elements
which accept the dir
attribute); this provides for backward
compatibility with MathML 2.0 which had no notion of directionality.
Also, it accepts the mathbackground
attribute in the same sense
as mstyle
and other presentation elements to set the background
color of the bounding box, rather than specifying a default for the attribute
(see Section 3.1.10 Mathematics style attributes common to presentation elements)
In addition to those attributes, the math
element accepts:
Name | values | default |
---|---|---|
display | "block" | "inline" | inline |
specifies whether the enclosed MathML expression should be rendered
as a separate vertical block (in display style)
or inline, aligned with adjacent text.
When display ="block", displaystyle is initialized
to "true",
whereas when display ="inline", displaystyle
is initialized to "false";
in both cases scriptlevel is initialized to 0
(See Section 3.1.6 Displaystyle and Scriptlevel).
Moreover, when the math element is embedded in a larger document,
a block math element should be treated as a block element as appropriate
for the document type (typically as a new vertical block),
whereas an inline math element should be treated as inline
(typically exactly as if it were a sequence of words in normal text).
In particular, this applies to spacing and linebreaking: for instance,
there should not be spaces or line breaks inserted between inline math
and any immediately following punctuation.
When the display attribute is missing, a rendering agent is free to initialize
as appropriate to the context.
|
||
maxwidth | length | available width |
specifies the maximum width to be used for linebreaking. The default is the maximum width available in the surrounding environment. If that value cannot be determined, the renderer should assume an infinite rendering width. | ||
overflow | "linebreak" | "scroll" | "elide" | "truncate" | "scale" | linebreak |
specifies the preferred handing in cases where an expression is too long to fit in the allowed width. See the discussion below. | ||
altimg | URI | none |
provides a URI referring to an image to display as a fall-back for user agents that do not support embedded MathML. | ||
altimg-width | length | width of altimg |
specifies the width to display altimg , scaling the image if necessary;
See altimg-height .
|
||
altimg-height | length | height of altimg |
specifies the height to display altimg , scaling the image if necessary;
if only one of the attributes altimg-width and altimg-height
are given, the scaling should preserve the image's aspect ratio;
if neither attribute is given, the image should be shown at its natural size.
|
||
altimg-valign | length | "top" | "middle" | "bottom" | 0ex |
specifies the vertical alignment of the image with respect to adjacent inline material.
A positive value of altimg-valign shifts the bottom of the image above the
current baseline, while a negative value lowers it.
The keyword "top" aligns the top of the image with the top of adjacent inline material;
"center" aligns the middle of the image to the middle of adjacent material;
"bottom" aligns the bottom of the image to the bottom of adjacent material
(not necessarily the baseline). This attribute only has effect
when display ="inline".
By default, the bottom of the image aligns to the baseline.
|
||
alttext | string | none |
provides a textual alternative as a fall-back for user agents that do not support embedded MathML or images. | ||
cdgroup | URI | none |
specifies a CD group file that acts as a catalogue of CD bases for locating
OpenMath content dictionaries of csymbol , annotation , and
annotation-xml elements in this math element; see Section 4.2.3 Content Symbols <csymbol> . When no cdgroup attribute is explicitly specified, the
document format embedding this math element may provide a method for determining
CD bases. Otherwise the system must determine a CD base; in the absence of specific
information http://www.openmath.org/cd is assumed as the CD base for all
csymbol , annotation , and annotation-xml elements. This is the
CD base for the collection of standard CDs maintained by the OpenMath Society.
|
In cases where size negotiation is not possible or fails
(for example in the case of an expression that is too long to fit in the allowed width),
the overflow
attribute is provided to suggest a processing method to the renderer.
Allowed values are:
Value | Meaning |
---|---|
"linebreak" | The expression will be broken across several lines. See Section 3.1.7 Linebreaking of Expressions for further discussion. |
"scroll" | The window provides a viewport into the larger complete display of the mathematical expression. Horizontal or vertical scroll bars are added to the window as necessary to allow the viewport to be moved to a different position. |
"elide" | The display is abbreviated by removing enough of it so that the remainder fits into the window. For example, a large polynomial might have the first and last terms displayed with "+ ... +" between them. Advanced renderers may provide a facility to zoom in on elided areas. |
"truncate" | The display is abbreviated by simply truncating it at the right and bottom borders. It is recommended that some indication of truncation is made to the viewer. |
"scale" | The fonts used to display the mathematical expression are chosen so that the full expression fits in the window. Note that this only happens if the expression is too large. In the case of a window larger than necessary, the expression is shown at its normal size within the larger window. |
The following attributes of math
are deprecated:
Name | values | default |
---|---|---|
macros | URI * | none |
intended to provide a way of pointing to external macro definition files. Macros are not part of the MathML specification. | ||
mode | "display" | "inline" | inline |
specified whether the enclosed MathML expression should be rendered in
a display style or an inline style.
This attribute is deprecated in
favor of the display attribute.
|
Information nowadays is commonly 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. In addition, there are increasing numbers of Web services, some of which offer technically based materials and activities. Thus, although MathML can be written by hand and read by humans, whether machine-aided or just with much concentration, the future of MathML is largely tied to the ability to process it with software tools.
There are many different kinds of MathML processors: editors for authoring MathML expressions, translators for converting to and from other encodings, validators for checking MathML expressions, computation engines that evaluate, manipulate, or compare MathML expressions, and rendering engines that produce visual, aural, or tactile representations of mathematical notation. What it means to support MathML varies widely between applications. For example, the issues that arise with a validating parser are very different from those for an equation editor.
This section gives guidelines that describe different types of MathML support and make clear the extent of MathML support in a given application. Developers, users, and reviewers are encouraged to use these guidelines in characterizing products. The intention behind these guidelines is to facilitate reuse by and interoperability of MathML applications by accurately setting out their capabilities in quantifiable terms.
The W3C Math Working Group maintains MathML Compliance Guidelines. Consult this document for future updates on conformance activities and resources.
A valid MathML expression is an XML construct determined by the MathML RelaxNG Schema together with the additional requirements given in this specification.
We shall use the phrase "a MathML processor" to mean any application that can accept or produce a valid MathML expression. A MathML processor that both accepts and produces valid MathML expressions may be able to "round-trip" MathML. Perhaps the simplest example of an application that might round-trip a MathML expression would be an editor that writes it to a new file without modifications.
Three forms of MathML conformance are specified:
A MathML-input-conformant processor must accept all valid MathML expressions; it should appropriately translate all MathML expressions into application-specific form allowing native application operations to be performed.
A MathML-output-conformant processor must generate valid MathML, appropriately representing all application-specific data.
A MathML-round-trip-conformant processor must preserve MathML equivalence. Two MathML expressions are "equivalent" if and only if both expressions have the same interpretation (as stated by the MathML Schema and specification) under any relevant circumstances, by any MathML processor. Equivalence on an element-by-element basis is discussed elsewhere in this document.
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 recommended rendering rules throughout Chapter 3 Presentation Markup. However, in general, developers are given wide latitude to interpret what kind of MathML implementation is meaningful for their own particular application.
To clarify the difference between conformance and interpretation of what is meaningful, consider some examples:
In order to be MathML-input-conformant, a validating parser needs only to accept expressions, and return "true" for expressions that are valid MathML. In particular, it need not render or interpret the MathML expressions at all.
A MathML computer-algebra interface based on content markup might choose to ignore all presentation markup. Provided the interface accepts all valid MathML expressions including those containing presentation markup, it would be technically correct to characterize the application as MathML-input-conformant.
An equation editor might have an internal data representation that makes it easy to export some equations as MathML but not others. If the editor exports the simple equations as valid MathML, and merely displays an error message to the effect that conversion failed for the others, it is still technically MathML-output-conformant.
As the previous examples show, to be useful, the concept of MathML conformance 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 conformance. 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-conformant 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-conformant processors, information about currently available tools to validate MathML is maintained at the W3C MathML Validator. Developers of MathML-output-conformant 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 3.0 contains a number of features of earlier MathML 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 current MathML conformance.
In order to be MathML-output-conformant, authoring tools may not generate MathML markup containing deprecated features.
In order to be MathML-input-conformant, rendering and reading tools must support deprecated features if they are to be in conformance with MathML 1.x or MathML 2.x. They do not have to support deprecated features to be considered in conformance with MathML 3.0. However, all tools are encouraged to support the old forms as much as possible.
In order to be MathML-round-trip-conformant, a processor need only preserve MathML equivalence on expressions containing no deprecated features.
MathML 3.0 defines three basic extension mechanisms: the mglyph
element provides a way of displaying glyphs for non-Unicode
characters, and glyph variants for existing Unicode characters; the
maction
element uses attributes from other namespaces to obtain
implementation-specific parameters; and content markup makes use of
the definitionURL
attribute, as well as
Content Dictionaries and the cd
attribute, to point to external
definitions of mathematical semantics.
These extension mechanisms are important because they provide a way
of encoding concepts that are beyond the scope of MathML 3.0 as presently
explicitly specified, which
allows MathML to be used for exploring new ideas not yet susceptible
to standardization. However, as new ideas take hold, they may become
part of future standards. For example, an emerging character that
must be represented by an mglyph
element today may be
assigned a Unicode code point in the future. At that time,
representing the character directly by its Unicode code point would be
preferable. This transition into Unicode has
already taken place for hundreds of characters used for mathematics.
Because the possibility of future obsolescence is inherent in the
use of extension mechanisms to facilitate the discussion of new ideas,
MathML can reasonably make
no conformance requirements concerning the use of
extension mechanisms, even when alternative standard markup is
available. For example, using an mglyph
element to represent
an 'x' is permitted. However, authors and implementers are
strongly encouraged to use standard markup whenever possible.
Similarly, maintainers of documents employing MathML 3.0 extension
mechanisms are encouraged to monitor relevant standards activity
(e.g., Unicode, OpenMath, etc.) and to update documents as more
standardized markup becomes available.
If a MathML-input-conformant 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-conformant 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 intended to allow for good presentation and content markup. However it is never possible to cover all users' needs for 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 XML DTD or in a Schema. Although it can be done using extensions of the standard DTD, say, some authors will wish to use non-standard attributes to take advantage of renderer-specific capabilities while remaining strictly in conformance with the standard DTD.
To allow this, the MathML 1.0 specification [MathML1]
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 next period of evolution of MathML the
development of a general XML namespace mechanism
seemed to make 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. The
other
attribute remains deprecated in MathML 3.0.
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 3.7.1 Bind Action to Sub-Expression
<maction>
),
that information should be passed in using the other
attribute:
<maction actiontype="highlight" other="color='#ff0000'"> expression </maction>
From MathML 2.0 onwards, a color
attribute from another namespace would be used:
<body xmlns: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.
This chapter specifies the "presentation" elements of MathML, which can be used to describe the layout structure of mathematical notation.
Presentation elements correspond to the "constructors" of traditional mathematical notation — that is, to the basic kinds of symbols and expression-building structures out of which any particular piece of traditional mathematical notation is built. Because of the importance of traditional visual notation, the descriptions of the notational constructs the elements represent are usually given here in visual terms. However, the elements are medium-independent in the sense that they have been designed to contain enough information for good spoken renderings as well. Some attributes of these elements may make sense only for visual media, but most attributes can be treated in an analogous way in audio as well (for example, by a correspondence between time duration and horizontal extent).
MathML presentation elements only suggest (i.e. do not require) specific ways of rendering in order to allow for medium-dependent rendering and for individual preferences of style. This specification describes suggested visual rendering rules in some detail, but a particular MathML renderer is free to use its own rules as long as its renderings are intelligible.
The presentation elements are meant to express the syntactic
structure of mathematical notation in much the same way as titles, sections,
and paragraphs capture the higher-level syntactic structure of a
textual document. Because of this, a single row of identifiers and operators
will often be represented by multiple nested mrow
elements rather than
a single mrow
. For example, "x + a /
b" typically is represented as:
<mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> / </mo> <mi> b </mi> </mrow> </mrow> |
Similarly, superscripts are attached to the full expression constituting their base rather than to the just preceding character. This structure permits better-quality rendering of mathematics, especially when details of the rendering environment, such as display widths, are not known ahead of time to the document author. It also greatly eases automatic interpretation of the represented mathematical structures.
Certain characters are used
to name identifiers or operators that in traditional notation render the
same as other symbols or usually rendered invisibly. For example, the entities
ⅆ
, ⅇ
, and
ⅈ
denote notational symbols semantically distinct from visually
identical letters used as simple variables.
Likewise, the entities
⁢
,
⁡
,
⁣
and the character U+2064
(INVISIBLE PLUS) usually render invisibly but represent significant information.
These entities have distinct spoken renderings, may influence visual linebreaking and spacing,
and may effect the evaluation or meaning of particular expressions.
Accordingly, authors should use these entities wherever they are applicable.
For instance, the expression represented visually as
"f(x)" would usually be spoken in English as
"f of x" rather than just
"f x". MathML conveys this meaning by using
the ⁡
operator after the
"f", which, in this case, can be aurally rendered as
"of".
The complete list of MathML entities is described in [Entities].
It is strongly recommended that, before reading the present chapter, one read Section 2.1 MathML Syntax and Grammar on MathML syntax and grammar, which contains important information on MathML notations and conventions. In particular, in this chapter it is assumed that the reader has an understanding of basic XML terminology described in Section 2.1.3 Children versus Arguments, and the attribute value notations and conventions described in Section 2.1.5 MathML Attribute Values.
The remainder of this section introduces MathML-specific terminology and conventions used in this chapter.
The presentation elements are divided into two classes. Token elements represent individual symbols, names, numbers, labels, etc. Layout schemata build expressions out of parts and can have only elements as content (except for whitespace, which they ignore). These are subdivided into General Layout, Script and Limit, Tabular Math and Elementary Math schemata. There are also a few empty elements used only in conjunction with certain layout schemata.
All individual "symbols" in a mathematical expression should be
represented by MathML token elements. The primary MathML token element
types are identifiers (e.g. variables or function names), numbers, and
operators (including fences, such as parentheses, and separators, such
as commas). There are also token elements used to represent text or
whitespace that has more aesthetic than mathematical significance
and other elements representing "string literals" for compatibility with
computer algebra systems. Note that although a token element
represents a single meaningful "symbol" (name, number, label,
mathematical symbol, etc.), such symbols may be comprised of more than
one character. For example sin
and 24
are
represented by the single tokens <mi>sin</mi>
and <mn>24</mn>
respectively.
In traditional mathematical notation, expressions are recursively constructed out of smaller expressions, and ultimately out of single symbols, with the parts grouped and positioned using one of a small set of notational structures, which can be thought of as "expression constructors". In MathML, expressions are constructed in the same way, with the layout schemata playing the role of the expression constructors. The layout schemata specify the way in which sub-expressions are built into larger expressions. The terminology derives from the fact that each layout schema corresponds to a different way of "laying out" its sub-expressions to form a larger expression in traditional mathematical typesetting.
The terminology used in this chapter for special classes of elements, and for relationships between elements, is as follows: The presentation elements are the MathML elements defined in this chapter. These elements are listed in Section 3.1.9 Summary of Presentation Elements. The content elements are the MathML elements defined in Chapter 4 Content Markup.
A MathML expression is a single instance of any of the
presentation elements with the exception of the empty elements
none
or mprescripts
,
or is a single instance of any of the content elements which are allowed as
content of presentation elements (described in Section 5.3.2 Content Markup in Presentation Markup).
A sub-expression of an expression
E is any MathML expression that is part of the content of
E, whether directly or indirectly,
i.e. whether it is a "child" of E or not.
Since layout schemata attach special meaning to the number and/or positions of their children, a child of a layout schema is also called an argument of that element. As a consequence of the above definitions, the content of a layout schema consists exactly of a sequence of zero or more elements that are its arguments.
Many of the elements described herein require a specific number of arguments (always 1, 2, or 3). In the detailed descriptions of element syntax given below, the number of required arguments is implicitly indicated by giving names for the arguments at various positions. A few elements have additional requirements on the number or type of arguments, which are described with the individual element. For example, some elements accept sequences of zero or more arguments — that is, they are allowed to occur with no arguments at all.
Note that MathML elements encoding rendered space do
count as arguments of the elements in which they appear.
See Section 3.2.7 Space <mspace/>
for a discussion of the proper use of such
space-like elements.
<mrow>
s
The elements listed in the following table as requiring 1*
argument (msqrt
, mstyle
, merror
,
mpadded
, mphantom
, menclose
,
mtd
, mscarry
,
and math
)
conceptually accept a single argument,
but actually accept any number of children.
If the number of children is 0 or is more than 1, they treat their contents
as a single inferred mrow
formed from all their children,
and treat this mrow
as the argument.
For example,
<mtd> </mtd>
is treated as if it were
<mtd> <mrow> </mrow> </mtd>
and
<msqrt> <mo> - </mo> <mn> 1 </mn> </msqrt> |
is treated as if it were
<msqrt> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> |
This feature allows MathML data not to contain (and its authors to
leave out) many mrow
elements that would otherwise be
necessary.
For convenience, here is a table of each element's argument count
requirements and the roles of individual arguments when these are
distinguished. An argument count of 1* indicates an inferred mrow
as described above.
Although the math
element is
not a presentation element, it is listed below for completeness.
Element | Required argument count | Argument roles (when these differ by position) |
---|---|---|
mrow |
0 or more | |
mfrac |
2 | numerator denominator |
msqrt |
1* | |
mroot |
2 | base index |
mstyle |
1* | |
merror |
1* | |
mpadded |
1* | |
mphantom |
1* | |
mfenced |
0 or more | |
menclose |
1* | |
msub |
2 | base subscript |
msup |
2 | base superscript |
msubsup |
3 | base subscript superscript |
munder |
2 | base underscript |
mover |
2 | base overscript |
munderover |
3 | base underscript overscript |
mmultiscripts |
1 or more | base
(subscript superscript)*
[<mprescripts/>
(presubscript presuperscript)*]
|
mtable |
0 or more rows | 0 or more mtr or mlabeledtr elements
|
mlabeledtr |
1 or more | a label and 0 or more mtd elements
|
mtr |
0 or more | 0 or more mtd elements
|
mtd |
1* | |
mstack |
0 or more | |
mlongdiv |
3 or more | divisor result dividend (msrow | msgroup | mscarries | msline)* |
msgroup |
0 or more | |
msrow |
0 or more | |
mscarries |
0 or more | |
mscarry |
1* | |
maction |
1 or more | depend on actiontype attribute
|
math |
1* |
Certain MathML presentation elements exhibit special behaviors in certain contexts. Such special behaviors are discussed in the detailed element descriptions below. However, for convenience, some of the most important classes of special behavior are listed here.
Certain elements are considered space-like; these are defined in
Section 3.2.7 Space <mspace/>
. This definition affects some of the suggested rendering
rules for mo
elements (Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
).
Certain elements, e.g. msup
, are able to
embellish operators that are their first argument. These elements are
listed in Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
, which precisely defines an "embellished
operator" and explains how this affects the suggested rendering rules
for stretchy operators.
In the notations familiar to most readers, both the overall layout and the textual symbols are arranged from left to right (LTR). Yet, as alluded to in the introduction, mathematics written in Hebrew or in locales such as Morocco or Persia, the overall layout is used unchanged, but the embedded symbols (often Hebrew or Arabic) are written right to left (RTL). Moreover, in most of the Arabic speaking world, the notation is arranged entirely RTL; thus a superscript is still raised, but it follows the base on the left rather than the right.
MathML 3.0 therefore recognizes two distinct directionalities: the directionality of the text and symbols within token elements and the overall directionality represented by Layout Schemata. These two facets are discussed below.
The overall directionality for a formula, basically
the direction of the Layout Schemata, is specified by
the dir
attribute on the containing math
element
(see Section 2.2 The Top-Level
<math>
Element).
The default is ltr
. When dir
="rtl"
is used, the layout is simply the mirror image of the conventional
European layout. That is, shifts up or down are unchanged,
but the progression in laying out is from right to left.
For example, in a RTL layout, sub- and superscripts appear to the left of the base; the surd for a root appears at the right, with the bar continuing over the base to the left. The layout details for elements whose behaviour depends on directionality are given in the discussion of the element. In those discussions, the terms leading and trailing are used to specify a side of an object when which side to use depends on the directionality; ie. leading means left in LTR but right in RTL. The terms left and right may otherwise be safely assumed to mean left and right.
The overall directionality is usually set on the math
, but
may also be switched for individual subformula by using the dir
attribute on mrow
or mstyle
elements.
When not specified, all elements inherit the directionality of their container.
The text directionality comes into play for the MathML token elements
that can contain text (mtext
, mo
, mi
, mn
and ms
) and is determined by the Unicode properties of that text.
A token element containing exclusively LTR or RTL characters
is displayed straightforwardly in the given direction.
When a mixture of directions is involved used, such as RTL Arabic
and LTR numbers, the Unicode bidirectional algorithm [Bidi]
is applied. This algorithm specifies how runs of characters
with the same direction are processed and how the runs are (re)ordered.
The base, or initial, direction is given by the overall directionality
described above (Section 3.1.5.1 Overall Directionality of Mathematics Formulas) and affects
how weakly directional characters are treated and how runs are nested.
(The dir
attribute is thus allowed on token elements to specify
the initial directionality that may be needed in rare cases.)
Any mglyph
or malignmark
elements appearing within
a token element are effectively neutral and have no effect
on ordering.
The important thing to notice is that the bidirectional algorithm is applied independently to the contents of each token element; each token element is an independent run of characters.
Other features of Unicode and scripts that should be respected are ‘mirroring’ and ‘glyph shaping’. Some Unicode characters are marked as being mirrored when presented in a RTL context; that is, the character is drawn as if it were mirrored or replaced by a corresponding character. Thus an opening parenthesis, ‘(’, in RTL will display as ‘)’. Conversely, the solidus (/ U+002F) is not marked as mirrored. Thus, an Arabic author that desires the slash to be reversed in an inline division should explicitly use reverse solidus (\ U+005C) or an alternative such as the mirroring DIVISION SLASH (U+2215).
Additionally, calligraphic scripts such as Arabic blend, or connect sequences of characters together, changing their appearance. As this can have an significant impact on readability, as well as aesthetics, it is important to apply such shaping if possible. Glyph shaping, like directionality, applies to each token element's contents individually.
Please note that for the transfinite cardinals represented by Hebrew characters, the code points U+2135-U+2138 (ALEF SYMBOL, BET SYMBOL, GIMEL SYMBOL, DALET SYMBOL) should be used. These are strong left-to-right.
So-called ‘displayed’ formulas, those appearing on a line by themselves,
typically make more generous use of vertical space than inline formulas,
which should blend into the adjacent text without intruding into
neighboring lines. For example, in a displayed summation, the limits
are placed above and below the summation symbol, while when it appears inline
the limits would appear in the sub and superscript position.
For similar reasons, sub- and superscripts,
nested fractions and other constructs typically display in a
smaller size than the main part of the formula.
MathML implicitly associates with every presentation node
a displaystyle
and scriptlevel
reflecting whether
a more expansive vertical layout applies and the level of scripting
in the current context.
These values are
initialized by the math
element
according to the display
attribute.
They are automatically adjusted by the
various script and limit schemata elements,
and the elements
mfrac
and
mroot
,
which typically set displaystyle
false and increment scriptlevel
for some or all of their arguments.
(See the description for each element for the specific rules used.)
They also may be set explicitly via the displaystyle
and scriptlevel
attributes on the mstyle
element
or the displaystyle
attribute of mtable
.
In all other cases, they are inherited from the node's parent.
The displaystyle
affects the amount of vertical space used to lay out a formula:
when true, the more spacious layout of displayed equations is used,
whereas when false a more compact layout of inline formula is used.
This primarily affects the interpretation
of the largeop
and movablelimits
attributes of
the mo
element.
However, more sophisticated renderers are free to use
this attribute to render more or less compactly.
The main effect of scriptlevel
is to control the font size.
Typically, the higher the scriptlevel
, the smaller the font size.
(Non-visual renderers can respond to the font size in an analogous way for their medium.)
Whenever the scriptlevel
is changed, whether automatically or explicitly,
the current font size is multiplied by the value of
scriptsizemultiplier
to the power of the change in scriptlevel
.
However, changes to the font size due to scriptlevel
changes should
never reduce the size below scriptminsize
to prevent scripts
becoming unreadably small.
The default scriptsizemultiplier
is approximately the square root of 1/2
whereas scriptminsize
defaults to 8 points;
these values may be changed on mstyle
; see Section 3.3.4 Style Change <mstyle>
.
Note that the scriptlevel
attribute of mstyle
allows arbitrary
values of scriptlevel
to be obtained, including negative values which
result in increased font sizes.
The changes to the font size due to scriptlevel
should be viewed
as being imposed from ‘outside’ the node.
This means that the effect of scriptlevel
is applied
before an explicit mathsize
(see Section 3.2.2 Mathematics style attributes common to token elements)
on a token child of mfrac
.
Thus, the mathsize
effectively overrides the effect of scriptlevel
.
However, that change to scriptlevel
changes the current font size,
which affects the meaning of an "em" length
(see Section 2.1.5.2 Length Valued Attributes)
and so the scriptlevel
still may have an effect in such cases.
Note also that since mathsize
is not constrained by scriptminsize
,
such direct changes to font size can result in scripts smaller than scriptminsize
.
Note that direct changes to current font size, whether by
CSS or by the mathsize
attribute (See Section 3.2.2 Mathematics style attributes common to token elements),
have no effect on the value of scriptlevel
.
TEX's \displaystyle, \textstyle, \scriptstyle, and \scriptscriptstyle
correspond to displaystyle
and scriptlevel
as
"true" and "0",
"false" and "0",
"false" and "1",
and "false" and "2", respectively.
Thus, math
's
display
="block" corresponds to \displaystyle,
while display
="inline" corresponds to \textstyle.
MathML provides support for both automatic and manual (forced)
linebreaking of expressions to break excessively long
expressions into several lines.
All such linebreaks take place within mrow
(including inferred mrow
; see Section 3.1.3.1 Inferred <mrow>
s)
or mfenced
.
The breaks typically take place at mo
elements
and also, for backwards compatibility, at mspace
.
Renderers may also choose to place automatic linebreaks at other points
such as between adjacent mi
elements or even within a token element
such as a very long mn
element. MathML does not provide a means to
specify such linebreaks, but if a render chooses to linebreak at such a point,
it should indent the following line according to the
indentation attributes
that are in effect at that point.
Automatic linebreaking occurs when the containing math
element
has overflow
="linebreak"
and the display engine determines that there is not enough space available to
display the entire formula. The available width must therefore be known
to the renderer. Like font properties, one is assumed to be inherited from the environment in
which the MathML element lives. If no width can be determined, an
infinite width should be assumed. Inside of a mtable
,
each column has some width. This width may be specified as an attribute
or determined by the contents. This width should be used as the
line wrapping width for linebreaking, and each entry in an mtable
is linewrapped as needed.
Forced linebreaks are specified by using
linebreak
="newline"
on a mo
or mspace
element.
Both automatic and manual linebreaking can occur within the same formula.
Automatic linebreaking of subexpressions of mfrac
, msqrt
, mroot
and menclose
and the various script elements is not required.
Renderers are free to ignore forced breaks within those elements if they choose.
Attributes on mo
and possibly on mspace
elements control
linebreaking and indentation of the following line. The aspects of linebreaking
that can be controlled are:
Where — attributes determine the desirability of
a linebreak at a specific operator or space, in particular whether a
break is required or inhibited. These can only be set on
mo
and mspace
elements.
(See Section 3.2.5.2.2 Linebreaking attributes.)
Operator Display/Position — when a linebreak occurs,
determines whether the operator will appear
at the end of the line, at the beginning of the next line, or in both positions;
and how much vertical space should be added after the linebreak.
These attributes can be set on mo
elements or inherited from
mstyle
or math
elements.
(See Section 3.2.5.2.2 Linebreaking attributes.)
Indentation — determines the indentation of the
line following a linebreak, including indenting so that the next line aligns
with some point in a previous line.
These attributes can be set on mo
elements or
inherited from mstyle
or math
elements.
(See Section 3.2.5.2.3 Indentation attributes.)
When a math element appears in an inline context, it may obey whatever paragraph flow rules
are employed by the document's text rendering engine.
Such rules are necessarily outside of the scope of this specification.
Alternatively, it may use the value of the math
element's overflow attribute.
(See Section 2.2.1 Attributes.)
One method of linebreaking that works reasonably well is sometimes referred to as a "best-fit" algorithm. It works by computing a "penalty" for each potential break point on a line. The break point with the smallest penalty is chosen and the algorithm then works on the next line. Three useful factors in a penalty calculation are:
How much of the line width (after subtracting of the indent) is unused? The more unused, the higher the penalty.
How deeply nested is the breakpoint in the expression tree? The expression
tree's depth is roughly similar to the nesting depth of mrow
s.
The more deeply nested the break point, the higher the penalty.
Does a linebreak here make layout of the next line difficult? If the next line is not the last line and if the indentingstyle uses information about the linebreak point to determine how much to indent, then the amount of room left for linebreaking on the next line must be considered; i.e., linebreaks that leave very little room to draw the next line result in a higher penalty.
Whether "linebreak" has been specified: "nobreak" effectively sets the penalty to infinity, "badbreak" increases the penalty "goodbreak" decreases the penalty, and "newline" effectively sets the penalty to 0.
This algorithm takes time proportional to the number of token elements times the number of lines.
A common method for breaking inline expressions that are too long for the space remaining on the current line is to pick an appropriate break point for the expression and place the expression up to that point on the current line and place the remainder of the expression on the following line. This can be done by:
Querying the text processing engine for the minimum and maximum amount of space available on the current line.
Using a variation of the automatic linebreaking algorithm
given above),
and/or using hints provided by linebreak attributes
on mo
or mspace
elements, to choose a line break.
The goal is that the first part of the formula fits "comfortably" on the current line
while breaking at a point that results in keeping related
parts of an expression on the same line.
The remainder of the formula begins on the next line, positioned both vertically and horizontally according to the paragraph flow; MathML's indentation attributes are ignored in this algorithm.
If the remainder does not fit on a line, steps 1 - 3 are repeated for the second and subsequent lines. Unlike the for the first line, some part of the expression must be placed these lines so that the algorithm terminates.
Some use-cases require precise control of the math layout and presentation. Several MathML elements and attributes expressly support such fine-tuning of the rendering. However, MathML rendering agents exhibit wide variability in their presentation of the the same MathML expression due to difference in platforms, font availability, and requirements particular to the agent itself (see Section 3.1 Introduction). The overuse of explicit rendering control may yield a ‘perfect’ layout on one platform, but give much worse presentation on others. The following sections clarify the kinds of problems that can occur.
For particular expressions, authors may be tempted to use the
mpadded
,
mspace
,
mphantom
, and
mtext
elements to improve
("tweak") the spacing generated by a specific renderer.
Without explicit spacing rules, various MathML renders may use different spacing algorithms. Consequently, different MathML renderers may position symbols in different locations relative to each other. Say that renderer B, for example, provides improved spacing for a particular expression over renderer A. Authors are strongly warned that "tweaking" the layout for renderer A may produce very poor results in renderer B, very likely worse than without any explicit adjustment at all.
Even when a specific choice of renderer can be assumed, its spacing rules may be improved in successive versions, so that the effect of tweaking in a given MathML document may grow worse with time. Also, when style sheet mechanisms are extended to MathML, even one version of a renderer may use different spacing rules for users with different style sheets.
Therefore, it is suggested that MathML markup never use
mpadded
or mspace
elements
to tweak the rendering of specific expressions, unless the MathML is
generated solely to be viewed using one specific version of one MathML
renderer, using one specific style sheet (if style sheets are
available in that renderer).
In cases where the temptation to improve spacing proves too strong,
careful use of mpadded
,
mphantom
, or the alignment elements (Section 3.5.5 Alignment Markers
<maligngroup/>
, <malignmark/>
) may give more portable results than the
direct insertion of extra space using mspace
or
mtext
. Advice given to the implementers of MathML
renderers might be still more productive, in the long run.
MathML elements that permit "negative spacing", namely
mspace
, mpadded
, and
mo
, could in theory be used to simulate new
notations or "overstruck" characters by the visual overlap of the
renderings of more than one MathML sub-expression.
This practice is strongly discouraged in all situations, for the following reasons:
it will give different results in different MathML renderers (so the warning about "tweaking" applies), especially if attempts are made to render glyphs outside the bounding box of the MathML expression;
it is likely to appear much worse than a more standard construct supported by good renderers;
such expressions are almost certain to be uninterpretable by audio renderers, computer algebra systems, text searches for standard symbols, or other processors of MathML input.
More generally, any construct that uses spacing to convey mathematical meaning, rather than simply as an aid to viewing expression structure, is discouraged. That is, the constructs that are discouraged are those that would be interpreted differently by a human viewer of rendered MathML if all explicit spacing was removed.
Consider using the mglyph
element
for cases such as this. If such spacing constructs are used in spite of this warning, they should
be enclosed in a semantics
element that also
provides an additional MathML expression that can be interpreted in a
standard way. See Section 5.1 Annotation Framework for further discussion.
The above warning also applies to most uses of rendering
attributes to alter the meaning conveyed by an expression, with the
exception of attributes on mi
(such as mathvariant
)
used to distinguish one variable from another.
mi |
identifier |
mn |
number |
mo |
operator, fence, or separator |
mtext |
text |
mspace |
space |
ms |
string literal |
Additionally, the mglyph
element
may be used within Token elements to represent non-standard symbols as images.
mrow |
group any number of sub-expressions horizontally |
mfrac |
form a fraction from two sub-expressions |
msqrt |
form a square root (radical without an index) |
mroot |
form a radical with specified index |
mstyle |
style change |
merror |
enclose a syntax error message from a preprocessor |
mpadded |
adjust space around content |
mphantom |
make content invisible but preserve its size |
mfenced |
surround content with a pair of fences |
menclose |
enclose content with a stretching symbol such as a long division sign. |
msub |
attach a subscript to a base |
msup |
attach a superscript to a base |
msubsup |
attach a subscript-superscript pair to a base |
munder |
attach an underscript to a base |
mover |
attach an overscript to a base |
munderover |
attach an underscript-overscript pair to a base |
mmultiscripts |
attach prescripts and tensor indices to a base |
mtable |
table or matrix |
mlabeledtr |
row in a table or matrix with a label or equation number |
mtr |
row in a table or matrix |
mtd |
one entry in a table or matrix |
maligngroup and
malignmark |
alignment markers |
mstack |
columns of aligned characters |
mlongdiv |
similar to msgroup, with the addition of a divisor and result |
msgroup |
a group of rows in an mstack that are shifted by similar amounts |
msrow |
a row in an mstack |
mscarries |
row in an mstack that whose contents represent carries or borrows |
mscarry |
one entry in an mscarries |
msline |
horizontal line inside of mstack |
maction |
bind actions to a sub-expression |
In addition to the attributes listed in Section 2.1.6 Attributes Shared by all MathML Elements, all MathML presentation elements accept the following two attributes:
Name | values | default |
---|---|---|
mathcolor | color | inherited |
Specifies the foreground color to use when drawing the components of this element,
such as the content for token elements or any lines, surds, or other decorations.
It also establishes the default mathcolor used for child elements
when used on a layout element.
|
||
mathbackground | color | "transparent" | transparent |
Specifies the background color to be used to fill in the bounding box of the element and its children. The default, "transparent", lets the background color, if any, used in the current rendering context to show through. |
These style attributes are primarily intended for visual media.
They are not expected to affect the intended semantics of displayed
expressions, but are for use in highlighting or drawing attention
to the affected subexpressions. For example, a red "x" is not assumed
to be semantically different than a black "x", in contrast to
variables with different mathvariant
(See Section 3.2.2 Mathematics style attributes common to token elements).
Since MathML expressions are often embedded in a textual data format such as HTML, the MathML renderer should inherit the foreground color used in the context in which the MathML appears. Note, however, that MathML doesn't specify the mechanism by which style information is inherited from the rendering environment. See Section 3.2.2 Mathematics style attributes common to token elements for more details.
Note that the suggested MathML visual rendering rules do not define the
precise extent of the region whose background is affected by the
mathbackground
attribute,
except that, when the content does not have
negative dimensions and its drawing region is not overlapped by other
drawing due to surrounding negative spacing, this region should lie
behind all the drawing done to render the content, but should not lie behind any of the
drawing done to render surrounding expressions. The effect of overlap
of drawing regions caused by negative spacing on the extent of the
region affected by the mathbackground
attribute is not
defined by these rules.
Token elements in presentation markup are broadly intended to represent the smallest units of mathematical notation which carry meaning. Tokens are roughly analogous to words in text. However, because of the precise, symbolic nature of mathematical notation, the various categories and properties of token elements figure prominently in MathML markup. By contrast, in textual data, individual words rarely need to be marked up or styled specially.
Frequently, tokens consist of a single character denoting a mathematical symbol. Other cases, e.g. function names, involve multi-character tokens. Further, because traditional mathematical notation makes wide use of symbols distinguished by their typographical properties (e.g. a Fraktur 'g' for a Lie algebra, or a bold 'x' for a vector), care must be taken to insure that styling mechanisms respect typographical properties which carry meaning. Consequently, characters, tokens, and typographical properties of symbols are closely related to one another in MathML.
Token elements represent
identifiers (mi
),
numbers (mn
),
operators (mo
),
text (mtext
),
strings (ms
)
and spacing (mspace
).
The mglyph
element
may be used within token elements
to represent non-standard symbols by images.
Preceding detailed discussion of the individual elements,
the next two subsections discuss the allowable content of
token elements and the attributes common to them.
<mglyph/>
Character data in MathML markup is only allowed to occur as part of
the content of token elements. Whitespace between elements is ignored.
With the exception of the empty mspace
element,
token elements can contain any sequence of zero or more Unicode characters,
or mglyph
or
malignmark
elements.
The mglyph
element is used
to represent non-standard characters or symbols by images;
the malignmark
element establishes an alignment point for use within
table constructs, and is otherwise invisible (See Section 3.5.5 Alignment Markers
<maligngroup/>
, <malignmark/>
).
Characters can be either represented directly as Unicode character data, or indirectly via numeric or character entity references. See Chapter 7 Characters, Entities and Fonts for a discussion of the advantages and disadvantages of numeric character references versus entity references, and [Entities] for a full list of the entity names available. Also, see Section 7.7 Anomalous Mathematical Characters for a discussion of the appropriate character content to choose for certain applications.
Token elements (other than mspace
) should
be rendered as their content, if any, (i.e. in the visual case, as a
closely-spaced horizontal row of standard glyphs for the characters
or images for the mglyph
s in their content).
An mspace
element is rendered as a blank space of a width determined by its attributes.
Rendering algorithms should also take into account the
mathematics style attributes as described below, and modify surrounding
spacing by rules or attributes specific to each type of token
element. The directional characteristics of the content must
also be respected (see Section 3.1.5.2 Bidirectional Layout in Token Elements).
A large class of mathematical symbols are single letter identifiers typically used as variable names in formulas. Different font variants of a letter are treated as separate symbols. For example, a Fraktur 'g' might denote a Lie algebra, while a Roman 'g' denotes the corresponding Lie group. These letter-like symbols are traditionally typeset differently than the same characters appearing in text, using different spacing and ligature conventions. These characters must also be treated specially by style mechanisms, since arbitrary style transformations can change meaning in an expression.
For these reasons, Unicode contains more than nine hundred Math Alphanumeric Symbol characters corresponding to letter-like symbols. These characters are in the Secondary Multilingual Plane (SMP). See [Entities] for more information. As valid Unicode data, these characters are permitted in MathML and, as tools and fonts for them become widely available, we anticipate they will be the predominant way of denoting letter-like symbols.
MathML also provides an alternative encoding
for these characters using only Basic Multilingual Plane
(BMP) characters together with markup. MathML defines a
correspondence between token elements with certain combinations of BMP
character data and the mathvariant
attribute and tokens
containing SMP Math Alphanumeric Symbol characters. Processing
applications that accept SMP characters are required to treat the
corresponding BMP and attribute combinations identically. This is particularly important for applications that
support searching and/or equality testing.
The mathvariant
attribute is described in more detail in
Section 3.2.2 Mathematics style attributes common to token elements,
and a complete technical description of the corresponding characters is given in
Section 7.5 Mathematical Alphanumeric Symbols.
<mglyph/>
The mglyph
element provides a mechanism
for displaying images to represent non-standard symbols.
It may be used within the content of the token elements
mi
, mn
, mo
, mtext
or ms
where existing Unicode characters are not adequate.
Unicode defines a large number of characters used in mathematics and, in most cases, glyphs representing these characters are widely available in a variety of fonts. Although these characters should meet almost all users needs, MathML recognizes that mathematics is not static and that new characters and symbols are added when convenient. Characters that become well accepted will likely be eventually incorporated by the Unicode Consortium or other standards bodies, but that is often a lengthy process.
Note that the glyph's src
attribute uniquely identifies the mglyph
;
two mglyph
s with the same values for src
should
be considered identical by applications that must determine whether
two characters/glyphs are identical.
The mglyph
element accepts the attributes listed in
Section 3.1.10 Mathematics style attributes common to presentation elements, but note that mathcolor
has no effect.
The background color, mathbackground
, should show through
if the specified image has transparency.
mglyph
also accepts the additional attributes listed here.
Name | values | default |
---|---|---|
src | URI | required |
Specifies the location of the image resource; it may be a URI relative to the base-URI of the source of the MathML, if any. | ||
width | length | from image |
Specifies the desired width of the glyph; see height .
|
||
height | length | from image |
Specifies the desired height of the glyph.
If only one of width and height are given,
the image should be scaled to preserve the aspect ratio;
if neither are given, the image should be displayed at its natural size.
|
||
valign | length | 0ex |
Specifies the baseline alignment point of the image with respect to the current baseline. A positive value shifts the bottom of the image above the current baseline while a negative value lowers it. A value of 0 (the default) means that the baseline of the image is at the bottom of the image. | ||
alt | string | required |
Provides an alternate name for the glyph. If the specified image can't be found or displayed, the renderer may use this name in a warning message or some unknown glyph notation. The name might also be used by an audio renderer or symbol processing system and should be chosen to be descriptive. |
Note that the src
and alt
attributes are required for correct usage in MathML 3, however this is not enforced by the schema due to the deprecated usage described below.
The following example illustrates how a researcher might use
the mglyph
construct with a set of images to work
with braid group notation.
<mrow> <mi><mglyph src="my-braid-23" alt="2 3 braid"/></mi> <mo>+</mo> <mi><mglyph src="my-braid-132" alt="1 3 2 braid"/></mi> <mo>=</mo> <mi><mglyph src="my-braid-13" alt="1 3 braid"/></mi> </mrow>
This might render as:
Originally, mglyph
was designed to provide access to non-standard
fonts. Since this functionality was seldom implemented, nor were downloadable
web fonts widely available, this use of mglyph
has been deprecated.
For reference, the following attributes were previously defined:
Name | values |
---|---|
fontfamily | string |
the name of a font that may be available to a MathML renderer, or a CSS font specification; See Section 6.5 Using CSS with MathML and CSS [CSS21] for more information. | |
index | integer |
Specified a position of the desired glyph within the font named
by the fontfamily attribute (see Section 3.2.2.1 Deprecated style attributes on token elements).
|
In MathML 1 and 2, both were required attributes; they are now optional
and should be ignored unless the src
attribute is missing.
Additionally, in MathML 2, mglyph
accepted the attributes described in
Section 3.2.2 Mathematics style attributes common to token elements
(mathvariant
and mathsize
, along with the attributes deprecated there);
to make clear that mglyph
is not a token element, and since
these attributes have no effect in any case, these attributes have been deprecated.
In addition to the attributes defined for all presentation elements
(Section 3.1.10 Mathematics style attributes common to presentation elements), MathML includes two mathematics style attributes
as well as a directionality attribute
valid on all presentation token elements,
as well as the math
and mstyle
elements;
dir
is also valid on mrow
elements.
The attributes are:
Name | values | default |
---|---|---|
mathvariant | "normal" | "bold" | "italic" | "bold-italic" | "double-struck" | "bold-fraktur" | "script" | "bold-script" | "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" | "sans-serif-bold-italic" | "monospace" | "initial" | "tailed" | "looped" | "stretched" | normal (except on <mi> )
|
Specifies the logical class of the token. Note that this class is more than styling, it typically conveys semantic intent; see the discussion below. | ||
mathsize | "small" | "normal" | "big" | length | inherited |
Specifies the size to display the token content. The values "small" and "big" choose a size smaller or larger than the current font size, but leave the exact proportions unspecified; "normal" is allowed for completeness, but since it is equivalent to "100%" or "1em", it has no effect. | ||
dir | "ltr" | "rtl" | inherited |
specifies the initial directionality for text within the token:
ltr (Left To Right) or rtl (Right To Left).
This attribute should only be needed in rare cases involving weak or neutral characters;
see Section 3.1.5.1 Overall Directionality of Mathematics Formulas for further discussion.
It has no effect on mspace .
|
The mathvariant
attribute defines logical classes of token
elements. Each class provides a collection of typographically-related
symbolic tokens. Each token has a specific meaning within a given
mathematical expression and, therefore, needs to be visually
distinguished and protected from inadvertent document-wide style
changes which might change its meaning. Each token is identified
by the combination of the mathvariant
attribute value
and the character data in the token element.
When MathML rendering takes place in an environment where CSS is
available, the mathematics style attributes can be viewed as
predefined selectors for CSS style rules.
See Section 6.5 Using CSS with MathML for discussion of the
interaction of MathML and CSS.
Also, see [MathMLforCSS] for discussion of rendering MathML by CSS
and a sample CSS style sheet.
When CSS is not available, it is up to the internal style mechanism of the rendering application
to visually distinguish the different logical classes.
Most MathML renderers will probably want to rely on some degree to additional,
internal style processing algorithms.
In particular, the mathvariant
attribute does not follow the CSS inheritance model;
the default value is "normal" (non-slanted)
for all tokens except for mi
with single-character content.
See Section 3.2.3 Identifier <mi>
for details.
Renderers have complete freedom in
mapping mathematics style attributes to specific rendering properties.
However, in practice, the mathematics style attribute names and values
suggest obvious typographical properties, and renderers should attempt
to respect these natural interpretations as far as possible. For
example, it is reasonable to render a token with the
mathvariant
attribute set to "sans-serif" in
Helvetica or Arial. However, rendering the token in a Times Roman
font could be seriously misleading and should be avoided.
In principle, any mathvariant
value may be used with any
character data to define a specific symbolic token. In practice,
only certain combinations of character data and mathvariant
values will be visually distinguished by a given renderer. For example,
there is no clear-cut rendering for a "fraktur alpha" or a "bold italic
Kanji" character, and the mathvariant
values "initial",
"tailed", "looped", and "stretched" are appropriate only for Arabic
characters.
Certain combinations of character data and mathvariant
values are equivalent to assigned Unicode code points that encode
mathematical alphanumeric symbols. These Unicode code points are
the ones in the
Arabic Mathematical Alphabetic Symbols block U+1EE00 to U+1EEFF,
Mathematical Alphanumeric Symbols block U+1D400 to U+1D7FF,
listed in the Unicode standard, and the ones in the
Letterlike
Symbols range U+2100 to U+214F that represent "holes" in the
alphabets in the SMP, listed in Section 7.5 Mathematical Alphanumeric Symbols.
These characters are described in detail in section 2.2 of
UTR #25.
The description of each such character in the Unicode standard
provides an unstyled character to which it would be equivalent
except for a font change that corresponds to a mathvariant
value. A token element that uses the unstyled character in combination
with the corresponding mathvariant
value is equivalent to a
token element that uses the mathematical alphanumeric symbol character
without the mathvariant
attribute. Note that the appearance
of a mathematical alphanumeric symbol character should not be altered
by surrounding mathvariant
or other style declarations.
Renderers should support those combinations of character data and
mathvariant
values that correspond to Unicode characters,
and that they can visually distinguish using available font characters.
Renderers may ignore or support those combinations of character data
and mathvariant
values that do not correspond to an assigned
Unicode code point, and authors should recognize that support for
mathematical symbols that do not correspond to assigned Unicode code
points may vary widely from one renderer to another.
Since MathML expressions are often embedded in a textual data format such as XHTML, the surrounding text and the MathML must share rendering attributes such as font size, so that the renderings will be compatible in style. For this reason, most attribute values affecting text rendering are inherited from the rendering environment, as shown in the "default" column in the table above. (In cases where the surrounding text and the MathML are being rendered by separate software, e.g. a browser and a plug-in, it is also important for the rendering environment to provide the MathML renderer with additional information, such as the baseline position of surrounding text, which is not specified by any MathML attributes.) Note, however, that MathML doesn't specify the mechanism by which style information is inherited from the rendering environment.
If the requested mathsize
of the current font is not available, the
renderer should approximate it in the manner likely to lead to the
most intelligible, highest quality rendering.
Note that many MathML elements automatically change the font size
in some of their children; see the discussion in Section 3.1.6 Displaystyle and Scriptlevel.
The MathML 1.01 style attributes listed below are deprecated in MathML 2 and 3. These attributes were aligned to CSS but, in rendering environments that support CSS, it is preferable to use CSS directly to control the rendering properties corresponding to these attributes, rather than the attributes themselves. However as explained above, direct manipulation of these rendering properties by whatever means should usually be avoided. As a general rule, whenever there is a conflict between these deprecated attributes and the corresponding attributes (Section 3.2.2 Mathematics style attributes common to token elements), the former attributes should be ignored.
The deprecated attributes are:
Name | values | default |
---|---|---|
fontfamily | string | inherited |
Should be the name of a font that may be available to a MathML renderer,
or a CSS font specification; See Section 6.5 Using CSS with MathML
and CSS [CSS21] for more information.
Deprecated in favor of mathvariant .
|
||
fontweight | "normal" | "bold" | inherited |
Specified the font weight for the token.
Deprecated in favor of mathvariant .
|
||
fontstyle | "normal" | "italic" | normal (except on <mi> )
|
Specified the font style to use for the token.
Deprecated in favor of mathvariant .
|
||
fontsize | length | inherited |
Specified the size for the token.
Deprecated in favor of mathsize .
|
||
color | color | inherited |
Specified the color for the token.
Deprecated in favor of mathcolor .
|
||
background | color | "transparent" | transparent |
Specified the background color to be used to fill in the bounding box
of the element and its children. Deprecated in favor of mathbackground .
|
MathML can be combined with other formats as described in
Section 6.4 Combining MathML and Other Formats.
The recommendation is to embed other formats in MathML by extending the MathML
schema to allow additional elements to be children of the mtext
element or
other leaf elements as appropriate to the role they serve in the expression
(see Section 3.2.6.4 Mixing text and mathematics).
The directionality, font size, and other font attributes should inherit from
those that would be used for characters of the containing leaf element
(see Section 3.2.2 Mathematics style attributes common to token elements).
Here is an example of embedding SVG inside of mtext in an HTML context:
<mtable> <mtr> <mtd> <mtext><input type="text" placeholder="what shape is this?"/></mtext> </mtd> </mtr> <mtr> <mtd> <mtext> <svg xmlns="http://www.w3.org/2000/svg" width="4cm" height="4cm" viewBox="0 0 400 400"> <rect x="1" y="1" width="398" height="398" style="fill:none; stroke:blue"/> <path d="M 100 100 L 300 100 L 200 300 z" style="fill:red; stroke:blue; stroke-width:3"/> </svg> </mtext> </mtd> </mtr> </mtable> |
<mi>
An mi
element represents a symbolic name or
arbitrary text that should be rendered as an identifier. Identifiers
can include variables, function names, and symbolic constants.
A typical graphical renderer would render an mi
element
as its content (See Section 3.2.1
Token Element Content Characters, <mglyph/>
),
with no extra spacing around it (except spacing associated with
neighboring elements).
Not all "mathematical identifiers" are represented by
mi
elements — for example, subscripted or primed
variables should be represented using msub
or
msup
respectively. Conversely, arbitrary text
playing the role of a "term" (such as an ellipsis in a summed series)
can be represented using an mi
element, as shown
in an example in Section 3.2.6.4 Mixing text and mathematics.
It should be stressed that mi
is a
presentation element, and as such, it only indicates that its content
should be rendered as an identifier. In the majority of cases, the
contents of an mi
will actually represent a
mathematical identifier such as a variable or function name. However,
as the preceding paragraph indicates, the correspondence between
notations that should render as identifiers and notations that are
actually intended to represent mathematical identifiers is not
perfect. For an element whose semantics is guaranteed to be that of an
identifier, see the description of ci
in
Chapter 4 Content Markup.
mi
elements accept the attributes listed in
Section 3.2.2 Mathematics style attributes common to token elements, but in one case with a different default value:
Name | values | default |
---|---|---|
mathvariant | "normal" | "bold" | "italic" | "bold-italic" | "double-struck" | "bold-fraktur" | "script" | "bold-script" | "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" | "sans-serif-bold-italic" | "monospace" | "initial" | "tailed" | "looped" | "stretched" | (depends on content; described below) |
Specifies the logical class of the token. The default is "normal" (non-slanted) unless the content is a single character, in which case it would be "italic". |
Note that the deprecated fontstyle
attribute
defaults in the same way as mathvariant
, depending on the content.
Note that for purposes of determining equivalences of Math
Alphanumeric Symbol
characters (See Section 7.5 Mathematical Alphanumeric Symbols and Section 3.2.1.1 Alphanumeric symbol characters)
the value of the mathvariant
attribute should be resolved first,
including the special defaulting behavior described above.
<mi> x </mi> <mi> D </mi> <mi> sin </mi> <mi mathvariant='script'> L </mi> <mi></mi> |
An mi
element with no content is allowed;
<mi></mi>
might, for example, be used by an
"expression editor" to represent a location in a MathML expression
which requires a "term" (according to conventional syntax for
mathematics) but does not yet contain one.
Identifiers include function names such as
"sin". Expressions such as "sin x"
should be written using the character U+2061
(which also has the entity names ⁡
and ⁡
) as shown below;
see also the discussion of invisible operators in Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
.
<mrow> <mi> sin </mi> <mo> ⁡<!--FUNCTION APPLICATION--> </mo> <mi> x </mi> </mrow> |
Miscellaneous text that should be treated as a "term" can also be
represented by an mi
element, as in:
<mrow> <mn> 1 </mn> <mo> + </mo> <mi> …<!--HORIZONTAL ELLIPSIS--> </mi> <mo> + </mo> <mi> n </mi> </mrow> |
When an mi
is used in such exceptional
situations, explicitly setting the mathvariant
attribute
may give better results than the default behavior of some
renderers.
The names of symbolic constants should be represented as
mi
elements:
<mi> π<!--GREEK SMALL LETTER PI--> </mi> <mi> ⅈ<!--DOUBLE-STRUCK ITALIC SMALL I--> </mi> <mi> ⅇ<!--DOUBLE-STRUCK ITALIC SMALL E--> </mi> |
<mn>
An mn
element represents a "numeric
literal" or other data that should be rendered as a numeric
literal. Generally speaking, a numeric literal is a sequence of digits,
perhaps including a decimal point, representing an unsigned integer or real
number.
A typical graphical renderer would render an mn
element as
its content (See Section 3.2.1
Token Element Content Characters, <mglyph/>
), with no extra spacing around them
(except spacing from neighboring elements such as mo
).
mn
elements are typically rendered in an unslanted font.
The mathematical concept of a "number" can be quite
subtle and involved, depending on the context. As a consequence, not all
mathematical numbers should be represented using mn
; examples of mathematical numbers that should be
represented differently are shown below, and include
complex numbers, ratios of numbers shown as fractions, and names of numeric
constants.
Conversely, since mn
is a presentation
element, there are a few situations where it may be desirable to include
arbitrary text in the content of an mn
that
should merely render as a numeric literal, even though that content
may not be unambiguously interpretable as a number according to any
particular standard encoding of numbers as character sequences. As a
general rule, however, the mn
element should be
reserved for situations where its content is actually intended to
represent a numeric quantity in some fashion. For an element whose
semantics are guaranteed to be that of a particular kind of
mathematical number, see the description of cn
in
Chapter 4 Content Markup.
mn
elements accept the attributes listed in Section 3.2.2 Mathematics style attributes common to token elements.
<mn> 2 </mn> <mn> 0.123 </mn> <mn> 1,000,000 </mn> <mn> 2.1e10 </mn> <mn> 0xFFEF </mn> <mn> MCMLXIX </mn> <mn> twenty one </mn> |
<mn>
alone
Many mathematical numbers should be represented using presentation
elements other than mn
alone; this includes
complex numbers, ratios of numbers shown as fractions, and
names of numeric constants. Examples of MathML representations of
such numbers include:
<mrow> <mn> 2 </mn> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢<!--INVISIBLE TIMES--> </mo> <mi> ⅈ<!--DOUBLE-STRUCK ITALIC SMALL I--> </mi> </mrow> </mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mi> π<!--GREEK SMALL LETTER PI--> </mi> <mi> ⅇ<!--DOUBLE-STRUCK ITALIC SMALL E--> </mi> |
<mo>
An mo
element represents an operator or
anything that should be rendered as an operator. In general, the
notational conventions for mathematical operators are quite
complicated, and therefore MathML provides a relatively sophisticated
mechanism for specifying the rendering behavior of an
mo
element. As a consequence, in MathML the list
of things that should "render as an operator" includes a number of
notations that are not mathematical operators in the ordinary
sense. Besides ordinary operators with infix, prefix, or postfix
forms, these include fence characters such as braces, parentheses, and
"absolute value" bars; separators
such as comma and semicolon; and
mathematical accents such as a bar or tilde over a symbol.
We will use the term "operator" in this chapter to refer to operators in this broad sense.
Typical graphical renderers show all mo
elements as the content (See Section 3.2.1
Token Element Content Characters, <mglyph/>
),
with additional spacing around the element determined by its attributes and
further described below.
Renderers without access to complete fonts for the MathML character
set may choose to render an mo
element as
not precisely the characters in its content in some cases. For example,
<mo> ≤ </mo>
might be rendered as
<=
to a terminal. However, as a general rule,
renderers should attempt to render the content of an
mo
element as literally as possible.
That is,
<mo> ≤ </mo>
and
<mo> <= </mo>
should render differently.
The first one should render as a single character
representing a less-than-or-equal-to sign, and the second one as the
two-character sequence <=
.
All operators, in the general sense used here,
are subject to essentially the same rendering
attributes and rules. Subtle distinctions in the
rendering of these classes of symbols,
when they exist, are supported using the Boolean attributes fence
,
separator
and accent
, which can be used to distinguish these cases.
A key feature of the mo
element is that its
default attribute values are set on a case-by-case basis from an
"operator dictionary" as explained below. In particular, default
values for fence
, separator
and
accent
can usually be found in the operator dictionary
and therefore need not be specified on each mo
element.
Note that some mathematical operators are represented not by mo
elements alone, but by mo
elements "embellished" with (for example) surrounding
superscripts; this is further described below. Conversely, as presentation
elements, mo
elements can contain arbitrary text,
even when that text has no standard interpretation as an operator; for an
example, see the discussion "Mixing text and mathematics" in
Section 3.2.6 Text <mtext>
. See also Chapter 4 Content Markup for
definitions of MathML content elements that are guaranteed to have the
semantics of specific mathematical operators.
Note also that linebreaking, as discussed in
Section 3.1.7 Linebreaking of Expressions, usually takes place at operators
(either before or after, depending on local conventions).
Thus, mo
accepts attributes to encode the desirability
of breaking at a particular operator, as well as attributes
describing the treatment of the operator and indentation in case
the a linebreak is made at that operator.
mo
elements accept
the attributes listed in Section 3.2.2 Mathematics style attributes common to token elements
and the additional attributes listed here.
Since the display of operators is so critical in mathematics,
the mo
element accepts a large number of attributes;
these are described in the next three subsections.
Most attributes get their default values from an enclosing
mstyle
element, math
element,
from the containing document,
or from the
Section 3.2.5.7.1 The operator dictionary.
When a value that is listed as "inherited" is not explicitly given on an
mo
, mstyle
element, math
element, or found in the operator
dictionary for a given mo
element, the default value shown in
parentheses is used.
Name | values | default |
---|---|---|
form | "prefix" | "infix" | "postfix" | set by position of operator in an mrow |
Specifies the role of the operator in the enclosing expression. This role and the operator content affect the lookup of the operator in the operator dictionary which affects the spacing and other default properties; see Section 3.2.5.7.2 Default value of the form attribute. | ||
fence | "true" | "false" | set by dictionary (false) |
Specifies whether the operator represents a ‘fence’, such as a parenthesis. This attribute generally has no direct effect on the visual rendering, but may be useful in specific cases, such as non-visual renderers. | ||
separator | "true" | "false" | set by dictionary (false) |
Specifies whether the operator represents a ‘separator’, or punctuation. This attribute generally has no direct effect on the visual rendering, but may be useful in specific cases, such as non-visual renderers. | ||
lspace | length | set by dictionary (thickmathspace) |
Specifies the leading space appearing before the operator; see Section 3.2.5.7.5 Spacing around an operator. (Note that before is on the right in a RTL context; see Section 3.1.5 Directionality). | ||
rspace | length | set by dictionary (thickmathspace) |
Specifies the trailing space appearing after the operator; see Section 3.2.5.7.5 Spacing around an operator. (Note that after is on the left in a RTL context; see Section 3.1.5 Directionality). | ||
stretchy | "true" | "false" | set by dictionary (false) |
Specifies whether the operator should stretch to the size of adjacent material; see Section 3.2.5.8 Stretching of operators, fences and accents. | ||
symmetric | "true" | "false" | set by dictionary (false) |
Specifies whether the operator should be kept symmetric around the math axis when stretchy. Note this property only applies to vertically stretched symbols. See Section 3.2.5.8 Stretching of operators, fences and accents. | ||
maxsize | length | "infinity" | set by dictionary (infinity) |
Specifies the maximum size of the operator when stretchy; see Section 3.2.5.8 Stretching of operators, fences and accents. Unitless or percentage values indicate a multiple of the reference size, being the size of the unstretched glyph. | ||
minsize | length | set by dictionary (100%) |
Specifies the minimum size of the operator when stretchy; see Section 3.2.5.8 Stretching of operators, fences and accents. Unitless or percentage values indicate a multiple of the reference size, being the size of the unstretched glyph. | ||
largeop | "true" | "false" | set by dictionary (false) |
Specifies whether the operator is considered a ‘large’ operator,
that is, whether it should be drawn larger than normal when
displaystyle ="true"
(similar to using TEX's \displaystyle).
Examples of large operators include ∫
and ∏ .
See Section 3.1.6 Displaystyle and Scriptlevel for more discussion.
|
||
movablelimits | "true" | "false" | set by dictionary (false) |
Specifies whether under- and overscripts attached to
this operator ‘move’ to the more compact sub- and superscript positions
when displaystyle is false.
Examples of operators that typically have movablelimits ="true"
are ∑ , ∏ , and lim.
See Section 3.1.6 Displaystyle and Scriptlevel for more discussion.
|
||
accent | "true" | "false" | set by dictionary (false) |
Specifies whether this operator should be treated as an accent (diacritical mark)
when used as an underscript or overscript;
see munder ,
mover
and munderover .
|
The following attributes affect when a linebreak does or does not occur, and the appearance of the linebreak when it does occur.
Name | values | default |
---|---|---|
linebreak | "auto" | "newline" | "nobreak" | "goodbreak" | "badbreak" | auto |
Specifies the desirability of a linebreak occurring at this operator: the default "auto" indicates the renderer should use its default linebreaking algorithm to determine whether to break; "newline" is used to force a linebreak; For automatic linebreaking, "nobreak" forbids a break; "goodbreak" suggests a good position; "badbreak" suggests a poor position. | ||
lineleading | length | inherited (100%) |
Specifies the amount of vertical space to use after a linebreak. For tall lines, it is often clearer to use more leading at linebreaks. Rendering agents are free to choose an appropriate default. | ||
linebreakstyle | "before" | "after" | "duplicate" | "infixlinebreakstyle" | set by dictionary (before) |
Specifies whether a linebreak occurs ‘before’ or ‘after’ the operator
when a linebreaks occur on this operator; or whether the operator is duplicated.
"before" causes the operator to appears at the beginning of the new line
(but possibly indented);
"after" causes it to appear at the end of the line before the break.
"duplicate" places the operator at both positions.
"infixlinebreakstyle" uses the value that has been specified for
infix operators; This value (one of "before",
"after" or "duplicate") can be specified by
the application or bound by mstyle
("before" corresponds to the most common style of linebreaking).
|
||
linebreakmultchar | string | inherited (⁢) |
Specifies the character used to make an ⁢ operator visible at a linebreak.
For example, linebreakmultchar ="·" would make the
multiplication visible as a center dot.
|
linebreak
values on adjacent mo
and mspace
elements do
not interact; linebreak
="nobreak" on a mo
does
not, in itself, inhibit a break on a preceding or following (possibly nested)
mo
or mspace
element and does not interact with the linebreakstyle
attribute value of the preceding or following mo
element.
It does prevent breaks from occurring on either side of the mo
element in all other situations.
The following attributes affect indentation of the lines making up a formula.
Primarily these attributes control the positioning of new lines following a linebreak,
whether automatic or manual. However, indentalignfirst
and indentshiftfirst
also control the positioning of single line formula without any linebreaks.
When these attributes appear on mo
or mspace
they apply if a linebreak occurs
at that element.
When they appear on mstyle
or math
elements, they determine
defaults for the style to be used for any linebreaks occurring within.
Note that except for cases where heavily marked-up manual linebreaking is desired,
many of these attributes are most useful when bound on an
mstyle
or math
element.
Note that since the rendering context, such as the available width and current font, is not always available to the author of the MathML, a render may ignore the values of these attributes if they result in a line in which the remaining width is too small to usefully display the expression or if they result in a line in which the remaining width exceeds the available linewrapping width.
Name | values | default |
---|---|---|
indentalign | "left" | "center" | "right" | "auto" | "id" | inherited (auto) |
Specifies the positioning of lines when linebreaking takes place within an mrow ;
see below for discussion of the attribute values.
|
||
indentshift | length | inherited (0) |
Specifies an additional indentation offset relative to the position determined
by indentalign .
When the value is a percentage value or number without unit,
the value is relative to the
horizontal space that a MathML renderer has available, this is the current target width as used for
linebreaking as specified in Section 3.1.7 Linebreaking of Expressions
|
||
indenttarget | idref | inherited (none) |
Specifies the id of another element
whose horizontal position determines the position of indented lines
when indentalign ="id".
Note that the identified element may be outside of the current
math element, allowing for inter-expression alignment,
or may be within invisible content such as mphantom ;
it must appear before being referenced, however.
This may lead to an id being unavailable to a given renderer
or in a position that does not allow for alignment.
In such cases, the indentalign should revert to "auto".
|
||
indentalignfirst | "left" | "center" | "right" | "auto" | "id" | "indentalign" | inherited (indentalign) |
Specifies the indentation style to use for the first line of a formula; the value "indentalign" (the default) means to indent the same way as used for the general line. | ||
indentshiftfirst | length | "indentshift" | inherited (indentshift) |
Specifies the offset to use for the first line of a formula;
the value "indentshift" (the default) means
to use the same offset as used for the general line.
Percentage values and numbers without unit are interpreted as described for indentshift
|
||
indentalignlast | "left" | "center" | "right" | "auto" | "id" | "indentalign" | inherited (indentalign) |
Specifies the indentation style to use for the last line when a linebreak
occurs within a given mrow ;
the value "indentalign" (the default) means
to indent the same way as used for the general line.
When there are exactly two lines, the value of this attribute should
be used for the second line in preference to indentalign .
|
||
indentshiftlast | length | "indentshift" | inherited (indentshift) |
Specifies the offset to use for the last line when a linebreak
occurs within a given mrow ;
the value "indentshift" (the default) means
to indent the same way as used for the general line.
When there are exactly two lines, the value of this attribute should
be used for the second line in preference to indentshift .
Percentage values and numbers without unit are interpreted as described for indentshift
|
The legal values of indentalign are:
Value | Meaning |
---|---|
left | Align the left side of the next line to the left side of the line wrapping width |
center | Align the center of the next line to the center of the line wrapping width |
right | Align the right side of the next line to the right side of the line wrapping width |
auto | (default) indent using the renderer's default indenting style; this may be a fixed amount or one that varies with the depth of the element in the mrow nesting or some other similar method. |
id | Align the left side of the next line to the left side of the element
referenced by the idref
(given by indenttarget );
if no such element exists, use "auto" as the indentalign value
|
<mo> + </mo> |
<mo> < </mo> |
<mo> ≤<!--LESS-THAN OR EQUAL TO--> </mo> |
<mo> <= </mo> |
<mo> ++ </mo> |
<mo> ∑<!--N-ARY SUMMATION--> </mo> |
<mo> .NOT. </mo> |
<mo> and </mo> |
<mo> ⁢<!--INVISIBLE TIMES--> </mo> |
<mo mathvariant='bold'> + </mo> |
Note that the mo
elements in these examples
don't need explicit fence
or separator
attributes,
since these can be found using the
operator dictionary as described below. Some of these examples could also
be encoded using the mfenced
element described in
Section 3.3.8 Expression Inside Pair of Fences
<mfenced>
.
(a+b)
<mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> |
[0,1)
<mrow> <mo> [ </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> |
f(x,y)
<mrow> <mi> f </mi> <mo> ⁡<!--FUNCTION APPLICATION--> </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> |
Certain operators that are "invisible" in traditional
mathematical notation should be represented using specific entity
references within mo
elements, rather than simply
by nothing. The characters used for these "invisible
operators" are:
Character | Entity name | Short name | Examples of use |
---|---|---|---|
U+2061 | ⁡ |
⁡ |
f(x) sin x |
U+2062 | ⁢ |
⁢ |
xy |
U+2063 | ⁣ |
⁣ |
m12 |
U+2064 | 2¾ |
The MathML representations of the examples in the above table are:
<mrow> <mi> f </mi> <mo> ⁡<!--FUNCTION APPLICATION--> </mo> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> |
<mrow> <mi> sin </mi> <mo> ⁡<!--FUNCTION APPLICATION--> </mo> <mi> x </mi> </mrow> |
<mrow> <mi> x </mi> <mo> ⁢<!--INVISIBLE TIMES--> </mo> <mi> y </mi> </mrow> |
<msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> ⁣<!--INVISIBLE SEPARATOR--> </mo> <mn> 2 </mn> </mrow> </msub> |
<mrow> <mn> 2 </mn> <mo> ⁤<!--INVISIBLE PLUS--> </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> |
The reasons for using specific mo
elements for
invisible operators include:
such operators should often have specific effects on visual
rendering (particularly spacing and linebreaking rules) that are not
the same as either the lack of any operator, or spacing represented by
mspace
or mtext
elements;
these operators should often have specific audio renderings different than that of the lack of any operator;
automatic semantic interpretation of MathML presentation elements is made easier by the explicit specification of such operators.
For example, an audio renderer might render f(x)
(represented as in the above examples) by speaking "f of x", but use
the word "times" in its rendering of xy.
Although its rendering must still be different depending on the structure
of neighboring elements (sometimes leaving out "of" or
"times" entirely), its task is made much easier by the use of
a different mo
element for each invisible
operator.
MathML also includes ⅆ
(U+2146) for use
in an mo
element representing the differential
operator symbol usually denoted by "d". The reasons for
explicitly using this special character are similar to those for using
the special characters for invisible operators described in the
preceding section.
<mo>
elements
Typical visual rendering behaviors for mo
elements are more complex than for the other MathML token elements, so
the rules for rendering them are described in this separate
subsection.
Note that, like all rendering rules in MathML, these rules are suggestions rather than requirements. Furthermore, no attempt is made to specify the rendering completely; rather, enough information is given to make the intended effect of the various rendering attributes as clear as possible.
Many mathematical symbols, such as an integral sign, a plus sign,
or a parenthesis, have a well-established, predictable, traditional
notational usage. Typically, this usage amounts to certain default
attribute values for mo
elements with specific
contents and a specific form
attribute. Since these
defaults vary from symbol to symbol, MathML anticipates that renderers
will have an "operator dictionary" of default attributes for
mo
elements (see Appendix C Operator Dictionary) indexed by each
mo
element's content and form
attribute. If an mo
element is not listed in the
dictionary, the default values shown in parentheses in the table of
attributes for mo
should be used, since these
values are typically acceptable for a generic operator.
Some operators are "overloaded", in the sense that they can occur
in more than one form (prefix, infix, or postfix), with possibly
different rendering properties for each form. For example, "+" can be
either a prefix or an infix operator. Typically, a visual renderer
would add space around both sides of an infix operator, while only in
front of a prefix operator. The form
attribute allows
specification of which form to use, in case more than one form is
possible according to the operator dictionary and the default value
described below is not suitable.
form
attribute
The form
attribute does not usually have to be
specified explicitly, since there are effective heuristic rules for
inferring the value of the form
attribute from the
context. If it is not specified, and there is more than one possible
form in the dictionary for an mo
element with
given content, the renderer should choose which form to use as follows
(but see the exception for embellished operators, described later):
If the operator is the first argument in an mrow
with more than one argument
(ignoring all space-like arguments (see Section 3.2.7 Space <mspace/>
) in the
determination of both the length and the first argument), the prefix form
is used;
if it is the last argument in an mrow
with more than one argument
(ignoring all space-like arguments), the postfix
form is used;
if it is the only element in an implicit or explicit mrow
and if it is in a script position of one of the elements listed in Section 3.4 Script and Limit Schemata,
the postfix form is used;
in all other cases, including when the operator is not part of an
mrow
, the infix form is used.
Note that the mrow
discussed above may be inferred;
See Section 3.1.3.1 Inferred <mrow>
s.
Opening fences should have form
="prefix"
,
and closing fences should have form
="postfix"
;
separators are usually "infix", but not always,
depending on their surroundings. As with ordinary operators,
these values do not usually need to be specified explicitly.
If the operator does not occur in the dictionary with the specified
form, the renderer should use one of the forms that is available
there, in the order of preference: infix, postfix, prefix; if no forms
are available for the given mo
element content, the
renderer should use the defaults given in parentheses in the table of
attributes for mo
.
There is one exception to the above rules for choosing an mo
element's default form
attribute. An mo
element that is
"embellished" by one or more nested subscripts, superscripts,
surrounding text or whitespace, or style changes behaves differently. It is
the embellished operator as a whole (this is defined precisely, below)
whose position in an mrow
is examined by the above
rules and whose surrounding spacing is affected by its form, not the mo
element at its core; however, the attributes
influencing this surrounding spacing are taken from the mo
element at the core (or from that element's
dictionary entry).
For example, the "+4" in
a+4b
should be considered an infix operator as a whole, due to its position
in the middle of an mrow
, but its rendering
attributes should be taken from the mo
element
representing the "+", or when those are not specified explicitly,
from the operator dictionary entry for <mo form="infix"> +
</mo>
.
The precise definition of an "embellished operator" is:
an mo
element;
or one of the elements
msub
,
msup
,
msubsup
,
munder
,
mover
,
munderover
,
mmultiscripts
,
mfrac
, or
semantics
(Section 5.1 Annotation Framework), whose first argument exists and is an embellished
operator;
or one of the elements
mstyle
,
mphantom
, or
mpadded
,
such that an mrow
containing the same
arguments would be an embellished operator;
or an maction
element whose selected
sub-expression exists and is an embellished operator;
or an mrow
whose arguments consist (in any order)
of one embellished operator and zero or more space-like elements.
Note that this definition permits nested embellishment only when there are no intervening enclosing elements not in the above list.
The above rules for choosing operator forms and defining
embellished operators are chosen so that in all ordinary cases it will
not be necessary for the author to specify a form
attribute.
The following notes are included as a rationale for certain aspects of the above definitions, but should not be important for most users of MathML.
An mfrac
is included as an
"embellisher" because of the common notation for a
differential operator:
<mfrac> <mo> ⅆ<!--DOUBLE-STRUCK ITALIC SMALL D--> </mo> <mrow> <mo> ⅆ<!--DOUBLE-STRUCK ITALIC SMALL D--> </mo> <mi> x </mi> </mrow> </mfrac> |
Since the definition of embellished operator affects the use of the
attributes related to stretching, it is important that it includes
embellished fences as well as ordinary operators; thus it applies to
any mo
element.
Note that an mrow
containing a single argument
is an embellished operator if and only if its argument is an embellished
operator. This is because an mrow
with a single
argument must be equivalent in all respects to that argument alone (as
discussed in Section 3.3.1 Horizontally Group Sub-Expressions
<mrow>
).
This means that an mo
element that is the sole argument of an mrow
will determine its default form
attribute based on that
mrow
's position in a surrounding, perhaps inferred,
mrow
(if there is one), rather than based on its own
position in the mrow
in which it is the sole argument.
Note that the above definition defines every
mo
element to be "embellished" — that is,
"embellished operator" can be considered (and implemented in
renderers) as a special class of MathML expressions, of which
mo
is a specific case.
The amount of horizontal space added around an operator (or embellished operator),
when it occurs in an mrow
, can be directly
specified by the lspace
and rspace
attributes. Note that lspace
and rspace
should
be interpreted as leading and trailing space, in the case of RTL direction.
By convention, operators that tend to bind tightly to their
arguments have smaller values for spacing than operators that tend to bind
less tightly. This convention should be followed in the operator dictionary
included with a MathML renderer.
Some renderers may choose to use no space around most operators appearing within subscripts or superscripts, as is done in TEX.
Non-graphical renderers should treat spacing attributes, and other rendering attributes described here, in analogous ways for their rendering medium. For example, more space might translate into a longer pause in an audio rendering.
Four attributes govern whether and how an operator (perhaps embellished)
stretches so that it matches the size of other elements: stretchy
, symmetric
, maxsize
, and minsize
. If an
operator has the attribute stretchy
="true", then it (that is, each character in its content)
obeys the stretching rules listed below, given the constraints imposed by
the fonts and font rendering system. In practice, typical renderers will
only be able to stretch a small set of characters, and quite possibly will
only be able to generate a discrete set of character sizes.
There is no provision in MathML for specifying in which direction
(horizontal or vertical) to stretch a specific character or operator;
rather, when stretchy
="true" it
should be stretched in each direction for which stretching is possible
and reasonable for that character.
It is up to the renderer to know in which directions it is reasonable to
stretch a character, if it can stretch the character.
Most characters can be stretched in at most one direction
by typical renderers, but some renderers may be able to stretch certain
characters, such as diagonal arrows, in both directions independently.
The minsize
and maxsize
attributes limit the amount of stretching (in either direction). These two
attributes are given as multipliers of the operator's normal size in the
direction or directions of stretching, or as absolute sizes using units.
For example, if a character has maxsize
="300%", then it
can grow to be no more than three times its normal (unstretched) size.
The symmetric
attribute governs whether the
height and
depth above and below the axis of the
character are forced to be equal
(by forcing both height and depth to become the maximum of the two).
An example of a situation where one might set
symmetric
="false"
arises with parentheses around a matrix not aligned on the axis, which
frequently occurs when multiplying non-square matrices. In this case, one
wants the parentheses to stretch to cover the matrix, whereas stretching
the parentheses symmetrically would cause them to protrude beyond one edge
of the matrix. The symmetric
attribute only applies
to characters that stretch vertically (otherwise it is ignored).
If a stretchy mo
element is embellished (as defined
earlier in this section), the mo
element at its core is
stretched to a size based on the context of the embellished operator
as a whole, i.e. to the same size as if the embellishments were not
present. For example, the parentheses in the following example (which
would typically be set to be stretchy by the operator dictionary) will be
stretched to the same size as each other, and the same size they would
have if they were not underlined and overlined, and furthermore will
cover the same vertical interval:
<mrow> <munder> <mo> ( </mo> <mo> _<!--LOW LINE--> </mo> </munder> <mfrac> <mi> a </mi> <mi> b </mi> </mfrac> <mover> <mo> ) </mo> <mo> ‾<!--OVERLINE--> </mo> </mover> </mrow> |
Note that this means that the stretching rules given below must
refer to the context of the embellished operator as a whole, not just
to the mo
element itself.
This shows one way to set the maximum size of a parenthesis so that
it does not grow, even though its default value is
stretchy
="true".
<mrow> <mo maxsize="100%"> ( </mo> <mfrac> <mi> a </mi> <mi> b </mi> </mfrac> <mo maxsize="100%"> ) </mo> </mrow> |
The above should render as
as opposed to the default rendering
.
Note that each parenthesis is sized independently; if only one of
them had maxsize
="100%", they would render with different
sizes.
The general rules governing stretchy operators are:
If a stretchy operator is a direct sub-expression of an mrow
element, or is the sole direct sub-expression of an
mtd
element in some row of a table, then it should
stretch to cover the height and depth (above and below the axis) of the non-stretchy direct sub-expressions in the
mrow
element or table row, unless stretching is
constrained by minsize
or maxsize
attributes.
In the case of an embellished stretchy operator, the preceding rule applies to the stretchy operator at its core.
The preceding rules also apply in situations where the mrow
element is inferred.
The rules for symmetric stretching only apply if
symmetric
="true" and if the stretching occurs in an mrow
or in an mtr
whose rowalign
value is either "baseline"
or "axis".
The following algorithm specifies the height and depth of vertically stretched characters:
Let maxheight
and maxdepth
be the maximum height and depth of the
non-stretchy siblings within the same mrow
or mtr
.
Let axis be the height of the math axis above the baseline.
Note that even if a minsize
or maxsize
value is set on a stretchy operator,
it is not used in the initial calculation of the maximum height and depth of an mrow
.
If symmetric
="true", then the computed height
and depth of the stretchy operator are:
height=max(maxheight-axis, maxdepth+axis) + axis depth =max(maxheight-axis, maxdepth+axis) - axis
Otherwise the height and depth are:
height= maxheight depth = maxdepth
If the total size = height+depth is less than minsize or greater than maxsize, increase or decrease both height and depth proportionately so that the effective size meets the constraint.
By default, most vertical arrows, along with most opening and closing fences are defined in the operator dictionary to stretch by default.
In the case of a stretchy operator in a table cell (i.e. within an
mtd
element), the above rules assume each cell of
the table row containing the stretchy operator covers exactly one row.
(Equivalently, the value of the rowspan
attribute is
assumed to be 1 for all the table cells in the table row, including
the cell containing the operator.) When this is not the case, the
operator should only be stretched vertically to cover those table
cells that are entirely within the set of table rows that the
operator's cell covers. Table cells that extend into rows not covered
by the stretchy operator's table cell should be ignored. See
Section 3.5.4.2 Attributes for details about the rowspan
attribute.
If a stretchy operator, or an embellished stretchy operator,
is a direct sub-expression of an munder
,
mover
, or munderover
element,
or if it is the sole direct sub-expression of an mtd
element in some
column of a table (see mtable
), then it, or the mo
element at its core, should stretch to cover
the width of the other direct sub-expressions in the given element (or
in the same table column), given the constraints mentioned above.
In the case of an embellished stretchy operator, the preceding rule applies to the stretchy operator at its core.
By default, most horizontal arrows and some accents stretch horizontally.
In the case of a stretchy operator in a table cell (i.e. within an
mtd
element), the above rules assume each cell of
the table column containing the stretchy operator covers exactly one
column. (Equivalently, the value of the columnspan
attribute is assumed to be 1 for all the table cells in the table row,
including the cell containing the operator.) When this is not the
case, the operator should only be stretched horizontally to cover
those table cells that are entirely within the set of table columns
that the operator's cell covers. Table cells that extend into columns
not covered by the stretchy operator's table cell should be
ignored. See Section 3.5.4.2 Attributes for details about the rowspan
attribute.
The rules for horizontal stretching include mtd
elements to allow arrows to stretch for use in commutative diagrams
laid out using mtable
. The rules for the horizontal
stretchiness include scripts to make examples such as the following
work:
<mrow> <mi> x </mi> <munder> <mo> →<!--RIGHTWARDS ARROW--> </mo> <mtext> maps to </mtext> </munder> <mi> y </mi> </mrow> |
This displays as
.
If a stretchy operator is not required to stretch (i.e. if it is
not in one of the locations mentioned above, or if there are no other
expressions whose size it should stretch to match), then it has the
standard (unstretched) size determined by the font and current
mathsize
.
If a stretchy operator is required to stretch, but all other expressions
in the containing element (as described above) are also stretchy,
all elements that can stretch should grow to the maximum of the normal
unstretched sizes of all elements in the containing object, if they can
grow that large. If the value of minsize
or maxsize
prevents
that, then the specified (min or max) size is
used.
For example, in an mrow
containing nothing but
vertically stretchy operators, each of the operators should stretch to
the maximum of all of their normal unstretched sizes, provided no
other attributes are set that override this behavior. Of course,
limitations in fonts or font rendering may result in the final,
stretched sizes being only approximately the same.
The following example demonstrates forced linebreaks and forced alignment:
<mrow> <mrow> <mi>f</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo id='eq1-equals'>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mn>4</mn> </msup> <mo linebreak='newline' linebreakstyle='before' indentalign='id' indenttarget='eq1-equals'>=</mo> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> <mo id='eq1-plus'>+</mo> <mrow> <mn>4</mn> <mo>⁢<!--INVISIBLE TIMES--></mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mrow> <mn>6</mn> <mo>⁢<!--INVISIBLE TIMES--></mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo linebreak='newline' linebreakstyle='before' indentalignlast='id' indenttarget='eq1-plus'>+</mo> <mrow> <mn>4</mn> <mo>⁢<!--INVISIBLE TIMES--></mo> <mi>x</mi> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </mrow> </mrow> |
This displays as
Note that because indentalignlast
defaults to "indentalign",
in the above example indentalign
could have been used in place of
indentalignlast
. Also, the specifying linebreakstyle='before'
is not needed because that is the default value.
<mtext>
An mtext
element is used to represent
arbitrary text that should be rendered as itself. In general, the
mtext
element is intended to denote commentary
text.
Note that some text with a clearly defined notational role might be
more appropriately marked up using mi
or
mo
; this is discussed further below.
An mtext
element can be used to contain
"renderable whitespace", i.e. invisible characters that are
intended to alter the positioning of surrounding elements. In non-graphical
media, such characters are intended to have an analogous effect, such as
introducing positive or negative time delays or affecting rhythm in an
audio renderer. This is not related to any whitespace in the source MathML
consisting of blanks, newlines, tabs, or carriage returns; whitespace
present directly in the source is trimmed and collapsed, as described in
Section 2.1.7 Collapsing Whitespace in Input. Whitespace that is intended to be rendered
as part of an element's content must be represented by entity references
or mspace
elements
(unless it consists only of single blanks between non-whitespace
characters).
mtext
elements accept the attributes listed in
Section 3.2.2 Mathematics style attributes common to token elements.
See also the warnings about the legal grouping of "space-like elements"
in Section 3.2.7 Space <mspace/>
, and about the use of
such elements for "tweaking" in Section 3.1.8 Warning about fine-tuning of presentation.
<mtext> Theorem 1: </mtext> <mtext>  <!--THIN SPACE--> </mtext> <mtext>   <!--space of width 5/18 em-->  <!--space of width 5/18 em--> </mtext> <mtext> /* a comment */ </mtext> |
In some cases, text embedded in mathematics could be more appropriately
represented using mo
or mi
elements.
For example, the expression 'there exists
such that f(x) <1' is equivalent to
and could be represented as:
<mrow> <mo> there exists </mo> <mrow> <mrow> <mi> δ<!--GREEK SMALL LETTER DELTA--> </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> such that </mo> <mrow> <mrow> <mi> f </mi> <mo> ⁡<!--FUNCTION APPLICATION--> </mo> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> |
An example involving an mi
element is:
x+x2+···+xn.
In this example, ellipsis should be represented using an mi
element, since it takes the place of a term in the
sum; (see Section 3.2.3 Identifier <mi>
).
On the other hand, expository text within MathML is best
represented with an mtext
element. An example
of this is:
Theorem 1: if x > 1, then x2 > x.
However, when MathML is embedded in HTML, or another document markup language, the example is probably best rendered with only the two inequalities represented as MathML at all, letting the text be part of the surrounding HTML.
Another factor to consider in deciding how to mark up text is the
effect on rendering. Text enclosed in an mo
element is unlikely to be found in a renderer's operator dictionary,
so it will be rendered with the format and spacing appropriate for an
"unrecognized operator", which may or may not be better than the
format and spacing for "text" obtained by using an
mtext
element. An ellipsis entity in an
mi
element is apt to be spaced more appropriately
for taking the place of a term within a series than if it appeared in
an mtext
element.
<mspace/>
An mspace
empty element represents a blank
space of any desired size, as set by its attributes. It can also be
used to make linebreaking suggestions to a visual renderer.
Note that the default values for attributes have been chosen so that
they typically will have no effect on rendering. Thus, the mspace
element is generally used with one
or more attribute values explicitly specified.
Note the warning about the legal grouping of "space-like
elements" given below, and the warning about the use of such
elements for "tweaking" in Section 3.1.8 Warning about fine-tuning of presentation.
See also the other elements that can render as
whitespace, namely mtext
, mphantom
, and
maligngroup
.
In addition to the attributes listed below,
mspace
elements accept the attributes described in Section 3.2.2 Mathematics style attributes common to token elements,
but note that mathvariant
and mathcolor
have no effect and that
mathsize
only affects the interpretation of units in sizing
attributes (see Section 2.1.5.2 Length Valued Attributes).
mspace
also accepts the indentation attributes described in Section 3.2.5.2.3 Indentation attributes.
Name | values | default |
---|---|---|
width | length | 0em |
Specifies the desired width of the space. | ||
height | length | 0ex |
Specifies the desired height (above the baseline) of the space. | ||
depth | length | 0ex |
Specifies the desired depth (below the baseline) of the space. | ||
linebreak | "auto" | "newline" | "nobreak" | "goodbreak" | "badbreak" | auto |
Specifies the desirability of a linebreak at this space. This attribute should be ignored if any dimensional attribute is set. |
Linebreaking was originally specified on mspace
in MathML2,
but controlling linebreaking on mo
is to be preferred starting with MathML 3.
MathML 3 adds new linebreaking attributes only to mo
, not mspace
.
However, because a linebreak can be specified on mspace
,
control over the indentation that follows that break can be specified using the attributes listed in
Section 3.2.5.2.3 Indentation attributes.
The value "indentingnewline" was defined in MathML2 for mspace
;
it is now deprecated. Its meaning is the same as newline
, which is compatible with its earlier use when no other linebreaking attributes are specified.
Note that linebreak
values on adjacent mo
and mspace
elements do
not interact; a "nobreak" on an mspace
will
not, in itself, inhibit a break on an adjacent mo
element.
<mspace height="3ex" depth="2ex"/> <mrow> <mi>a</mi> <mo id="firstop">+</mo> <mi>b</mi> <mspace linebreak="newline" indentalign="id" indenttarget="firstop"/> <mo>+</mo> <mi>c</mi> </mrow>
In the last example, mspace
will cause the line to end after the "b" and the following line to be indented so that the "+" that follows will align with
the "+" with id
="firstop".
A number of MathML presentation elements are "space-like" in the
sense that they typically render as whitespace, and do not affect the
mathematical meaning of the expressions in which they appear. As a
consequence, these elements often function in somewhat exceptional
ways in other MathML expressions. For example, space-like elements are
handled specially in the suggested rendering rules for
mo
given in Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
.
The following MathML elements are defined to be "space-like":
an mtext
, mspace
,
maligngroup
, or malignmark
element;
an mstyle
, mphantom
, or
mpadded
element, all of whose direct sub-expressions
are space-like;
an maction
element whose selected
sub-expression exists and is space-like;
an mrow
all of whose direct
sub-expressions are space-like.
Note that an mphantom
is not
automatically defined to be space-like, unless its content is
space-like. This is because operator spacing is affected by whether
adjacent elements are space-like. Since the
mphantom
element is primarily intended as an aid
in aligning expressions, operators adjacent to an
mphantom
should behave as if they were adjacent
to the contents of the mphantom
,
rather than to an equivalently sized area of whitespace.
Authors who insert space-like elements or
mphantom
elements into an existing MathML
expression should note that such elements are counted as
arguments, in elements that require a specific number of arguments,
or that interpret different argument positions differently.
Therefore, space-like elements inserted into such a MathML element
should be grouped with a neighboring argument of that element by
introducing an mrow
for that purpose. For example,
to allow for vertical alignment on the right edge of the base of a
superscript, the expression
<msup> <mi> x </mi> <malignmark edge="right"/> <mn> 2 </mn> </msup>
is illegal, because msup
must have exactly 2 arguments;
the correct expression would be:
<msup> <mrow> <mi> x </mi> <malignmark edge="right"/> </mrow> <mn> 2 </mn> </msup> |
See also the warning about "tweaking" in Section 3.1.8 Warning about fine-tuning of presentation.
<ms>
The ms
element is used to represent
"string literals" in expressions meant to be interpreted by
computer algebra systems or other systems containing "programming
languages". By default, string literals are displayed surrounded by
double quotes, with no extra spacing added around the string.
As explained in Section 3.2.6 Text <mtext>
, ordinary text
embedded in a mathematical expression should be marked up with mtext
,
or in some cases mo
or mi
, but never with ms
.
Note that the string literals encoded by ms
are made up of characters, mglyph
s and
malignmark
s rather than "ASCII
strings". For
example, <ms>&</ms>
represents a string
literal containing a single character, &
, and
<ms>&amp;</ms>
represents a string literal
containing 5 characters, the first one of which is
&
.
The content of ms
elements should be rendered with visible
"escaping" of certain characters in the content,
including at least the left and right quoting
characters, and preferably whitespace other than individual
space characters. The intent is for the viewer to see that the
expression is a string literal, and to see exactly which characters
form its content. For example, <ms>double quote is
"</ms>
might be rendered as "double quote is \"".
Like all token elements, ms
does trim and
collapse whitespace in its content according to the rules of
Section 2.1.7 Collapsing Whitespace in Input, so whitespace intended to remain in
the content should be encoded as described in that section.
ms
elements accept the attributes listed in
Section 3.2.2 Mathematics style attributes common to token elements, and additionally:
Name | values | default |
---|---|---|
lquote | string | " |
Specifies the opening quote to enclose the content. (not necessarily ‘left quote’ in RTL context). | ||
rquote | string | " |
Specifies the closing quote to enclose the content. (not necessarily ‘right quote’ in RTL context). |
Besides tokens there are several families of MathML presentation elements. One family of elements deals with various "scripting" notations, such as subscript and superscript. Another family is concerned with matrices and tables. The remainder of the elements, discussed in this section, describe other basic notations such as fractions and radicals, or deal with general functions such as setting style properties and error handling.
<mrow>
An mrow
element is used to group together any
number of sub-expressions, usually consisting of one or more mo
elements acting as "operators" on one
or more other expressions that are their "operands".
Several elements automatically treat their arguments as if they were
contained in an mrow
element. See the discussion of
inferred mrow
s in Section 3.1.3 Required Arguments.
See also mfenced
(Section 3.3.8 Expression Inside Pair of Fences
<mfenced>
),
which can effectively form an mrow
containing its arguments separated by commas.
mrow
elements are typically rendered visually
as a horizontal row of their arguments, left to right in the order in
which the arguments occur within a context with LTR directionality,
or right to left within a context with RTL directionality.
The dir
attribute can be used to specify
the directionality for a specific mrow
, otherwise it inherits the
directionality from the context. For aural agents, the arguments would be
rendered audibly as a sequence of renderings of
the arguments. The description in Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
of suggested rendering
rules for mo
elements assumes that all horizontal
spacing between operators and their operands is added by the rendering
of mo
elements (or, more generally, embellished
operators), not by the rendering of the mrow
s
they are contained in.
MathML provides support for both automatic and manual
linebreaking of expressions (that is, to break excessively long
expressions into several lines). All such linebreaks take place
within mrow
s, whether they are explicitly marked up
in the document, or inferred (See Section 3.1.3.1 Inferred <mrow>
s),
although the control of linebreaking is effected through attributes
on other elements (See Section 3.1.7 Linebreaking of Expressions).
mrow
elements accept the attribute listed below in addition to
those listed in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
dir | "ltr" | "rtl" | inherited |
specifies the overall directionality ltr (Left To Right) or
rtl (Right To Left) to use to layout the children of the row.
See Section 3.1.5.1 Overall Directionality of Mathematics Formulas for further discussion.
|
<mrow>
Sub-expressions should be grouped by the document author in the same way
as they are grouped in the mathematical interpretation of the expression;
that is, according to the underlying "syntax tree" of the
expression. Specifically, operators and their mathematical arguments should
occur in a single mrow
; more than one operator
should occur directly in one mrow
only when they
can be considered (in a syntactic sense) to act together on the interleaved
arguments, e.g. for a single parenthesized term and its parentheses, for
chains of relational operators, or for sequences of terms separated by
+
and -
. A precise rule is given below.
Proper grouping has several purposes: it improves display by possibly affecting spacing; it allows for more intelligent linebreaking and indentation; and it simplifies possible semantic interpretation of presentation elements by computer algebra systems, and audio renderers.
Although improper grouping will sometimes result in suboptimal
renderings, and will often make interpretation other than pure visual
rendering difficult or impossible, any grouping of expressions using
mrow
is allowed in MathML syntax; that is,
renderers should not assume the rules for proper grouping will be
followed.
<mrow>
of one argument
MathML renderers are required to treat an mrow
element containing exactly one argument as equivalent in all ways to
the single argument occurring alone, provided there are no attributes
on the mrow
element. If there are
attributes on the mrow
element, no
requirement of equivalence is imposed. This equivalence condition is
intended to simplify the implementation of MathML-generating software
such as template-based authoring tools. It directly affects the
definitions of embellished operator and space-like element and the
rules for determining the default value of the form
attribute of an mo
element;
see Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
and Section 3.2.7 Space <mspace/>
. See also the discussion of equivalence of MathML
expressions in Section 2.3 Conformance.
A precise rule for when and how to nest sub-expressions using
mrow
is especially desirable when generating
MathML automatically by conversion from other formats for displayed
mathematics, such as TEX, which don't always specify how sub-expressions
nest. When a precise rule for grouping is desired, the following rule
should be used:
Two adjacent operators, possibly embellished, possibly separated by operands (i.e. anything
other than operators), should occur in the same
mrow
only when the leading operator has an infix or
prefix form (perhaps inferred), the following operator has an infix or
postfix form, and the operators have the same priority in the
operator dictionary (Appendix C Operator Dictionary).
In all other cases, nested mrow
s should be used.
When forming a nested mrow
(during generation
of MathML) that includes just one of two successive operators with
the forms mentioned above (which mean that either operator could in
principle act on the intervening operand or operands), it is necessary
to decide which operator acts on those operands directly (or would do
so, if they were present). Ideally, this should be determined from the
original expression; for example, in conversion from an
operator-precedence-based format, it would be the operator with the
higher precedence.
Note that the above rule has no effect on whether any MathML expression is valid, only on the recommended way of generating MathML from other formats for displayed mathematics or directly from written notation.
(Some of the terminology used in stating the above rule in defined
in Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
.)
As an example, 2x+y-z should be written as:
<mrow> <mrow> <mn> 2 </mn> <mo> ⁢<!--INVISIBLE TIMES--> </mo> <mi> x </mi> </mrow> <mo> + </mo> <mi> y </mi> <mo> - </mo> <mi> z </mi> </mrow> |
The proper encoding of (x, y) furnishes a less obvious
example of nesting mrow
s:
<mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> |
In this case, a nested mrow
is required inside
the parentheses, since parentheses and commas, thought of as fence and
separator "operators", do not act together on their arguments.
<mfrac>
The mfrac
element is used for fractions. It can
also be used to mark up fraction-like objects such as binomial coefficients
and Legendre symbols. The syntax for mfrac
is
<mfrac> numerator denominator </mfrac>
The mfrac
element sets displaystyle
to "false", or if it
was already false increments scriptlevel
by 1,
within numerator and denominator.
(See Section 3.1.6 Displaystyle and Scriptlevel.)
mfrac
elements accept the attributes listed below
in addition to those listed in Section 3.1.10 Mathematics style attributes common to presentation elements.
The fraction line, if any, should be drawn using the color specified by mathcolor
.
Name | values | default |
---|---|---|
linethickness | length | "thin" | "medium" | "thick" | medium |
Specifies the thickness of the horizontal "fraction bar", or "rule" The default value is "medium", "thin" is thinner, but visible, "thick" is thicker; the exact thickness of these is left up to the rendering agent. | ||
numalign | "left" | "center" | "right" | center |
Specifies the alignment of the numerator over the fraction. | ||
denomalign | "left" | "center" | "right" | center |
Specifies the alignment of the denominator under the fraction. | ||
bevelled | "true" | "false" | false |
Specifies whether the fraction should be displayed in a beveled style (the numerator slightly raised, the denominator slightly lowered and both separated by a slash), rather than "build up" vertically. See below for an example. |
Thicker lines (e.g. linethickness
="thick") might be used with nested fractions;
a value of "0" renders without the bar such as for binomial coefficients.
These cases are shown below:
An example illustrating the bevelled form is shown below:
In a RTL directionality context, the numerator leads (on the right),
the denominator follows (on the left) and the diagonal line slants upwards going from right to left (See Section 3.1.5.1 Overall Directionality of Mathematics Formulas for clarification).
Although this format is an established convention, it is not universally
followed; for situations where a forward slash is desired in a RTL context,
alternative markup, such as an mo
within an mrow
should be used.
The examples shown above can be represented in MathML as:
<mrow> <mo> ( </mo> <mfrac linethickness="0"> <mi> a </mi> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> <mfrac linethickness="200%"> <mfrac> <mi> a </mi> <mi> b </mi> </mfrac> <mfrac> <mi> c </mi> <mi> d </mi> </mfrac> </mfrac> |
<mfrac> <mn> 1 </mn> <mrow> <msup> <mi> x </mi> <mn> 3 </mn> </msup> <mo> + </mo> <mfrac> <mi> x </mi> <mn> 3 </mn> </mfrac> </mrow> </mfrac> <mo> = </mo> <mfrac bevelled="true"> <mn> 1 </mn> <mrow> <msup> <mi> x </mi> <mn> 3 </mn> </msup> <mo> + </mo> <mfrac> <mi> x </mi> <mn> 3 </mn> </mfrac> </mrow> </mfrac> |
A more generic example is:
<mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mn> 2 </mn> </mfrac> |
<msqrt>
, <mroot>
These elements construct radicals. The msqrt
element is
used for square roots, while the mroot
element is used
to draw radicals with indices, e.g. a cube root. The syntax for these
elements is:
<msqrt> base </msqrt> <mroot> base index </mroot>
The mroot
element requires exactly 2 arguments.
However, msqrt
accepts a single argument, possibly
being an inferred mrow
of multiple children; see Section 3.1.3 Required Arguments.
The mroot
element increments scriptlevel
by 2,
and sets displaystyle
to "false", within
index, but leaves both attributes unchanged within base.
The msqrt
element leaves both
attributes unchanged within its argument.
(See Section 3.1.6 Displaystyle and Scriptlevel.)
Note that in a RTL directionality, the surd begins
on the right, rather than the left, along with the index in the case
of mroot
.
msqrt
and mroot
elements accept the attributes listed in
Section 3.1.10 Mathematics style attributes common to presentation elements. The surd and overbar should be drawn using the
color specified by mathcolor
.
<mstyle>
The mstyle
element is used to make style
changes that affect the rendering of its
contents.
Firstly, as a presentation element, it accepts
the attributes described in Section 3.1.10 Mathematics style attributes common to presentation elements.
Additionally, it
can be given any attribute
accepted by any other presentation element, except for the
attributes described below.
Finally,
the mstyle
element can be given certain special
attributes listed in the next subsection.
The mstyle
element accepts a single argument,
possibly being an inferred mrow
of multiple children;
see Section 3.1.3 Required Arguments.
Loosely speaking, the effect of the mstyle
element
is to change the default value of an attribute for the elements it
contains. Style changes work in one of several ways, depending on
the way in which default values are specified for an attribute.
The cases are:
Some attributes, such as displaystyle
or
scriptlevel
(explained below), are inherited
from the surrounding context when they are not explicitly set. Specifying
such an attribute on an mstyle
element sets the
value that will be inherited by its child elements. Unless a child element
overrides this inherited value, it will pass it on to its children, and
they will pass it to their children, and so on. But if a child element does
override it, either by an explicit attribute setting or automatically (as
is common for scriptlevel
), the new (overriding)
value will be passed on to that element's children, and then to their
children, etc, unless it is again overridden.
Other attributes, such as linethickness
on
mfrac
, have default values that are not normally
inherited. That is, if the linethickness
attribute
is not set on the mfrac
element,
it will normally use the default value of "1", even if it was
contained in a larger mfrac
element that set this
attribute to a different value. For attributes like this, specifying a
value with an mstyle
element has the effect of
changing the default value for all elements within its scope. The net
effect is that setting the attribute value with mstyle
propagates the change to all the elements it
contains directly or indirectly, except for the individual elements on
which the value is overridden. Unlike in the case of inherited attributes,
elements that explicitly override this attribute have no effect on this
attribute's value in their children.
Another group of attributes, such as stretchy
and form
, are
computed from operator dictionary information, position in the
enclosing mrow
, and other similar data. For
these attributes, a value specified by an enclosing mstyle
overrides the value that would normally be
computed.
Note that attribute values inherited from an
mstyle
in any manner affect a descendant element
in the mstyle
's content only if that attribute is
not given a value by the descendant element. On any element for
which the attribute is set explicitly, the value specified overrides the inherited value. The only exception to this
rule is when the attribute value
is documented as
specifying an incremental change to the value inherited from that
element's context or rendering environment.
Note also that the difference between inherited and non-inherited
attributes set by mstyle
, explained above, only
matters when the attribute is set on some element within the
mstyle
's contents that has descendants also
setting it. Thus it never matters for attributes, such as
mathsize
, which can only be set on token elements (or on
mstyle
itself).
MathML specifies that when
the attributes height
, depth
or width
are specified on an mstyle
element, they apply only to
mspace
elements, and not to the corresponding attributes of
mglyph
, mpadded
, or mtable
. Similarly, when
rowalign
, columnalign
, or groupalign
are specified on an mstyle
element, they apply only to the
mtable
element, and not the mtr
, mlabeledtr
,
mtd
, and maligngroup
elements.
When the lspace
attribute is set with mstyle
, it
applies only to the mo
element and not to mpadded
.
To be consistent, the voffset
attribute of the
mpadded
element can not be set on mstyle
.
When the deprecated fontfamily
attribute is specified on an
mstyle
element, it does not apply to the mglyph
element.
The deprecated index
attribute cannot be set on mstyle
.
When the align
attribute is set with mstyle
, it
applies only to the munder
, mover
, and munderover
elements, and not to the mtable
and mstack
elements.
The required attributes src
and alt
on mglyph
,
and actiontype
on maction
, cannot be set on mstyle
.
As a presentation element, mstyle
directly accepts
the mathcolor
and mathbackground
attributes.
Thus, the mathbackground
specifies the color to fill the bounding
box of the mstyle
element itself; it does not
specify the default background color.
This is an incompatible change from MathML 2, but we feel it is more useful
and intuitive. Since the default for mathcolor
is inherited,
this is no change in its behaviour.
As stated above, mstyle
accepts all
attributes of all MathML presentation elements which do not have
required values. That is, all attributes which have an explicit
default value or a default value which is inherited or computed are
accepted by the mstyle
element.
mstyle
elements accept the attributes listed in
Section 3.1.10 Mathematics style attributes common to presentation elements.
Additionally, mstyle
can be given the following special
attributes that are implicitly inherited by every MathML element as
part of its rendering environment:
Name | values | default |
---|---|---|
scriptlevel | ( "+" | "-" )? unsigned-integer | inherited |
Changes the scriptlevel in effect for the children.
When the value is given without a sign, it sets scriptlevel to the specified value;
when a sign is given, it increments ("+") or decrements ("-") the current value.
(Note that large decrements can result in negative values of scriptlevel ,
but these values are considered legal.)
See Section 3.1.6 Displaystyle and Scriptlevel.
|
||
displaystyle | "true" | "false" | inherited |
Changes the displaystyle in effect for the children.
See Section 3.1.6 Displaystyle and Scriptlevel.
|
||
scriptsizemultiplier | number | 0.71 |
Specifies the multiplier to be used to adjust font size due
to changes in scriptlevel .
See Section 3.1.6 Displaystyle and Scriptlevel.
|
||
scriptminsize | length | 8pt |
Specifies the minimum font size allowed due to changes in scriptlevel .
Note that this does not limit the font size due to changes to mathsize .
See Section 3.1.6 Displaystyle and Scriptlevel.
|
||
infixlinebreakstyle | "before" | "after" | "duplicate" | before |
Specifies the default linebreakstyle to use for infix operators; see Section 3.2.5.2.2 Linebreaking attributes | ||
decimalpoint | character | . |
specifies the character used to determine the alignment point within
mstack
and
mtable columns
when the "decimalpoint" value is used to specify the alignment.
The default, ".", is the decimal separator used to separate the integral
and decimal fractional parts of floating point numbers in many countries.
(See Section 3.6 Elementary Math and Section 3.5.5 Alignment Markers
<maligngroup/> , <malignmark/> ).
|
If scriptlevel
is changed incrementally by an
mstyle
element that also sets certain other
attributes, the overall effect of the changes may depend on the order
in which they are processed. In such cases, the attributes in the
following list should be processed in the following order, regardless
of the order in which they occur in the XML-format attribute list of
the mstyle
start tag:
scriptsizemultiplier
, scriptminsize
,
scriptlevel
, mathsize
.
MathML2 allowed the binding of namedspaces
to new values.
It appears that this capability was never implemented, and is now deprecated;
namedspaces are now considered constants.
For backwards compatibility, the following attributes are accepted
on the mstyle
element, but are expected to have no effect.
Name | values | default |
---|---|---|
veryverythinmathspace | length | 0.0555556em |
verythinmathspace | length | 0.111111em |
thinmathspace | length | 0.166667em |
mediummathspace | length | 0.222222em |
thickmathspace | length | 0.277778em |
verythickmathspace | length | 0.333333em |
veryverythickmathspace | length | 0.388889em |
The example of limiting the stretchiness of a parenthesis shown in the section on <mo>,
<mrow> <mo maxsize="100%"> ( </mo> <mfrac> <mi> a </mi> <mi> b </mi> </mfrac> <mo maxsize="100%"> ) </mo> </mrow> |
can be rewritten using mstyle
as:
<mstyle maxsize="100%"> <mrow> <mo> ( </mo> <mfrac> <mi> a </mi> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> </mstyle> |
<merror>
The merror
element displays its contents as an
"error message". This might be done, for example, by displaying the
contents in red, flashing the contents, or changing the background
color. The contents can be any expression or expression sequence.
merror
accepts
a single argument possibly being an inferred mrow
of multiple children;
see Section 3.1.3 Required Arguments.
The intent of this element is to provide a standard way for programs that generate MathML from other input to report syntax errors in their input. Since it is anticipated that preprocessors that parse input syntaxes designed for easy hand entry will be developed to generate MathML, it is important that they have the ability to indicate that a syntax error occurred at a certain point. See Section 2.3.2 Handling of Errors.
The suggested use of merror
for reporting
syntax errors is for a preprocessor to replace the erroneous part of
its input with an merror
element containing a
description of the error, while processing the surrounding expressions
normally as far as possible. By this means, the error message will be
rendered where the erroneous input would have appeared, had it been
correct; this makes it easier for an author to determine from the
rendered output what portion of the input was in error.
No specific error message format is suggested here, but as with
error messages from any program, the format should be designed to make
as clear as possible (to a human viewer of the rendered error message)
what was wrong with the input and how it can be fixed. If the
erroneous input contains correctly formatted subsections, it may be
useful for these to be preprocessed normally and included in the error
message (within the contents of the merror
element), taking advantage of the ability of
merror
to contain arbitrary MathML expressions
rather than only text.
merror
elements accept the attributes listed in
Section 3.1.10 Mathematics style attributes common to presentation elements.
If a MathML syntax-checking preprocessor received the input
<mfraction> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mn> 2 </mn> </mfraction>
which contains the non-MathML element mfraction
(presumably in place of the MathML element mfrac
),
it might generate the error message
<merror> <mtext> Unrecognized element: mfraction; arguments were: </mtext> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mtext> and </mtext> <mn> 2 </mn> </merror> |
Note that the preprocessor's input is not, in this case, valid MathML, but the error message it outputs is valid MathML.
<mpadded>
An mpadded
element renders the same as its child content,
but with the size of the child's bounding box and the relative positioning
point of its content modified according to
mpadded
's attributes. It
does not rescale (stretch or shrink) its content. The name of the
element reflects the typical use of mpadded
to add padding,
or extra space, around its content. However, mpadded
can be
used to make more general adjustments of size and positioning, and some
combinations, e.g. negative padding, can cause the content of
mpadded
to overlap the rendering of neighboring content. See
Section 3.1.8 Warning about fine-tuning of presentation for warnings about several
potential pitfalls of this effect.
The mpadded
element accepts
a single argument which may be an inferred mrow
of multiple children;
see Section 3.1.3 Required Arguments.
It is suggested that audio renderers add (or shorten) time delays
based on the attributes representing horizontal space
(width
and lspace
).
mpadded
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
height | ( "+" | "-" )? unsigned-number ( ("%" pseudo-unit?) | pseudo-unit | unit | namedspace )? | same as content |
Sets or increments the height of the mpadded element.
See below for discussion.
|
||
depth | ( "+" | "-" )? unsigned-number (("%" pseudo-unit?) | pseudo-unit | unit | namedspace )? | same as content |
Sets or increments the depth of the mpadded element.
See below for discussion.
|
||
width | ( "+" | "-" )? unsigned-number ( ("%" pseudo-unit?) | pseudo-unit | unit | namedspace )? | same as content |
Sets or increments the width of the mpadded element.
See below for discussion.
|
||
lspace | ( "+" | "-" )? unsigned-number ( ("%" pseudo-unit?) | pseudo-unit | unit | namedspace )? | 0em |
Sets the horizontal position of the child content. See below for discussion. | ||
voffset | ( "+" | "-" )? unsigned-number ( ("%" pseudo-unit?) | pseudo-unit | unit | namedspace )? | 0em |
Sets the vertical position of the child content. See below for discussion. |
The pseudo-unit syntax symbol is described below.
Also, height
, depth
and
width
attributes are
referred to as size attributes, while lspace
and voffset
attributes
are position attributes.
These attributes specify the size of the bounding box of the mpadded
element relative to the size of the bounding box of its child content, and specify
the position of the child content of the mpadded
element relative to the
natural positioning of the mpadded
element. The typographical
layout parameters determined by these attributes are described in the next subsection.
Depending on the form of the attribute value, a dimension may be set to a new value,
or specified relative to the child content's corresponding dimension. Values may be given as
multiples or percentages of any of the
dimensions of the normal rendering of the child content using so-called pseudo-units,
or they can be set directly using standard units Section 2.1.5.2 Length Valued Attributes.
If the value of a size attribute begins with a +
or -
sign,
it specifies an increment or decrement to the corresponding
dimension by the following length value. Otherwise the corresponding
dimension is set directly to the following length value.
Note that since a leading minus sign indicates a decrement, the size
attributes (height
, depth
, width
)
cannot be set directly to negative values. In addition, specifying a
decrement that would produce a net negative value for these attributes
has the same effect as
setting the attribute to zero. In other words, the effective
bounding box of an mpadded
element always has non-negative
dimensions. However, negative values are allowed for the relative positioning
attributes lspace
and voffset
.
Length values (excluding any sign) can be specified in several formats.
Each format begins with an unsigned-number,
which may be followed by
a %
sign (effectively scaling the number)
and an optional pseudo-unit,
by a pseudo-unit alone,
or by a unit (excepting %
).
The possible pseudo-units are the keywords height
,
depth
, and width
. They represent the length of the same-named dimension of the
mpadded
element's child content.
For any of these length formats, the resulting length
is the product of the number (possibly including the %
)
and the following pseudo-unit,
unit,
namedspace
or the default value for the attribute if no such unit or space is given.
Some examples of attribute formats using pseudo-units (explicit or
default) are as follows: depth="100%height"
and
depth="1.0height"
both set the depth of the
mpadded
element to the height of its content.
depth="105%"
sets the depth to 1.05 times the content's
depth, and either depth="+100%"
or
depth="200%"
sets the depth to twice the content's
depth.
The rules given above imply that all of the following attribute settings have the same effect, which is to leave the content's dimensions unchanged:
<mpadded width="+0em"> ... </mpadded> <mpadded width="+0%"> ... </mpadded> <mpadded width="-0em"> ... </mpadded> <mpadded width="-0height"> ... </mpadded> <mpadded width="100%"> ... </mpadded> <mpadded width="100%width"> ... </mpadded> <mpadded width="1width"> ... </mpadded> <mpadded width="1.0width"> ... </mpadded> <mpadded> ... </mpadded>
Note that the examples in the Version 2 of the MathML specification showed spaces within the attribute values, suggesting that this was the intended format. Formally, spaces are not allowed within these values, but implementers may wish to ignore such spaces to maximize backward compatibility.
See Appendix D Glossary for definitions of some of the typesetting terms used here.
The content of an mpadded
element defines a fragment of mathematical
notation, such as a character, fraction, or expression, that can be regarded as
a single typographical element with a natural positioning point relative to its
natural bounding box.
The size of the bounding box of an mpadded
element is
defined as the size of the bounding box of its content, except as
modified by the mpadded
element's
height
, depth
, and
width
attributes. The natural positioning point of the
child content of the mpadded
element is located to coincide
with the natural positioning point of the mpadded
element,
except as modified by the lspace
and voffset
attributes. Thus, the size attributes of mpadded
can be used
to expand or shrink the apparent bounding box of its content, and the
position attributes of mpadded
can be used to move the
content relative to the bounding box (and hence also neighboring elements).
Note that MathML doesn't define the precise relationship between "ink",
bounding boxes and positioning points, which are implementation
specific. Thus, absolute values for mpadded attributes may not be
portable between implementations.
The height
attribute specifies the vertical extent of the
bounding box of the mpadded
element above its baseline.
Increasing the height
increases the space between the baseline
of the mpadded
element and the content above it, and introduces
padding above the rendering of the child content. Decreasing the
height
reduces the space between the baseline of the
mpadded
element and the content above it, and removes
space above the rendering of the child content. Decreasing the
height
may cause content above the mpadded
element to overlap the rendering of the child content, and should
generally be avoided.
The depth
attribute specifies the vertical extent of the
bounding box of the mpadded
element below its baseline.
Increasing the depth
increases the space between the baseline
of the mpadded
element and the content below it, and introduces
padding below the rendering of the child content. Decreasing the
depth
reduces the space between the baseline of the mpadded
element and the content below it, and removes space below the rendering
of the child content. Decreasing the depth
may cause content
below the mpadded
element to overlap the rendering of the child
content, and should generally be avoided.
The width
attribute specifies the horizontal distance
between the positioning point of the mpadded
element and the
positioning point of the following content.
Increasing the width
increases the space between the
positioning point of the mpadded
element and the content
that follows it, and introduces padding after the rendering of the
child content. Decreasing the width
reduces the space
between the positioning point of the mpadded
element and
the content that follows it, and removes space after the rendering
of the child content. Setting the width
to zero causes
following content to be positioned at the positioning point of the
mpadded
element. Decreasing the width
should
generally be avoided, as it may cause overprinting of the following
content.
The lspace
attribute ("leading" space;
see Section 3.1.5.1 Overall Directionality of Mathematics Formulas) specifies the horizontal
location of the positioning point of the child content with respect to
the positioning point of the mpadded
element. By default they
coincide, and therefore absolute values for lspace have the same effect
as relative values.
Positive values for the lspace
attribute increase the space
between the preceding content and the child content, and introduce padding
before the rendering of the child content. Negative values for the
lspace
attributes reduce the space between the preceding
content and the child content, and may cause overprinting of the
preceding content, and should generally be avoided. Note that the
lspace
attribute does not affect the width
of
the mpadded
element, and so the lspace
attribute
will also affect the space between the child content and following
content, and may cause overprinting of the following content, unless
the width
is adjusted accordingly.
The voffset
attribute specifies the vertical location
of the positioning point of the child content with respect to the
positioning point of the mpadded
element. Positive values
for the voffset
attribute raise the rendering of the child
content above the baseline. Negative values for the voffset
attribute lower the rendering of the child content below the baseline.
In either case, the voffset
attribute may cause overprinting
of neighboring content, which should generally be avoided. Note that t
he voffset
attribute does not affect the height
or depth
of the mpadded
element, and so the voffset
attribute will also affect the space between the child content and neighboring
content, and may cause overprinting of the neighboring content, unless the
height
or depth
is adjusted accordingly.
MathML renderers should ensure that, except for the effects of the
attributes, the relative spacing between the contents of the
mpadded
element and surrounding MathML elements would
not be modified by replacing an mpadded
element with an
mrow
element with the same content, even if linebreaking
occurs within the mpadded
element. MathML does not define
how non-default attribute values of an mpadded
element interact
with the linebreaking algorithm.
The effects of the size and position attributes are illustrated
below. The following diagram illustrates the use of lspace
and voffset
to shift the position of child content without
modifying the mpadded
bounding box.
The corresponding MathML is:
<mrow> <mi>x</mi> <mpadded lspace="0.2em" voffset="0.3ex"> <mi>y</mi> </mpadded> <mi>z</mi> </mrow> |
The next diagram illustrates the use of
width
, height
and depth
to modifying the mpadded
bounding box without changing the relative position
of the child content.
The corresponding MathML is:
<mrow> <mi>x</mi> <mpadded width="+90%width" height="+0.3ex" depth="+0.3ex"> <mi>y</mi> </mpadded> <mi>z</mi> </mrow> |
The final diagram illustrates the generic use of mpadded
to modify both
the bounding box and relative position of child content.
The corresponding MathML is:
<mrow> <mi>x</mi> <mpadded lspace="0.3em" width="+0.6em"> <mi>y</mi> </mpadded> <mi>z</mi> </mrow> |
<mphantom>
The mphantom
element renders invisibly, but
with the same size and other dimensions, including baseline position,
that its contents would have if they were rendered
normally. mphantom
can be used to align parts of
an expression by invisibly duplicating sub-expressions.
The mphantom
element accepts
a single argument possibly being an inferred mrow
of multiple children;
see Section 3.1.3 Required Arguments.
Note that it is possible to wrap both an
mphantom
and an mpadded
element around one MathML expression, as in
<mphantom><mpadded attribute-settings>
... </mpadded></mphantom>
, to change its size and make it
invisible at the same time.
MathML renderers should ensure that the relative spacing between
the contents of an mphantom
element and the
surrounding MathML elements is the same as it would be if the
mphantom
element were replaced by an
mrow
element with the same content. This holds
even if linebreaking occurs within the mphantom
element.
For the above reason, mphantom
is
not considered space-like (Section 3.2.7 Space <mspace/>
) unless its
content is space-like, since the suggested rendering rules for
operators are affected by whether nearby elements are space-like. Even
so, the warning about the legal grouping of space-like elements may
apply to uses of mphantom
.
mphantom
elements accept the attributes listed in
Section 3.1.10 Mathematics style attributes common to presentation elements (the mathcolor
has no effect).
There is one situation where the preceding rules for rendering an
mphantom
may not give the desired effect. When an
mphantom
is wrapped around a subsequence of the
arguments of an mrow
, the default determination
of the form
attribute for an mo
element within the subsequence can change. (See the default value of
the form
attribute described in Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
.) It may be
necessary to add an explicit form
attribute to such an
mo
in these cases. This is illustrated in the
following example.
In this example, mphantom
is used to ensure
alignment of corresponding parts of the numerator and denominator of a
fraction:
<mfrac> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> x </mi> <mphantom> <mo form="infix"> + </mo> <mi> y </mi> </mphantom> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> |
This would render as something like
rather than as
The explicit attribute setting form
="infix"
on the
mo
element inside the mphantom
sets the
form
attribute to what it would have been in the absence of the
surrounding mphantom
. This is necessary since
otherwise, the +
sign would be interpreted as a prefix
operator, which might have slightly different spacing.
Alternatively, this problem could be avoided without any explicit
attribute settings, by wrapping each of the arguments
<mo>+</mo>
and <mi>y</mi>
in its
own mphantom
element, i.e.
<mfrac> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> x </mi> <mphantom> <mo> + </mo> </mphantom> <mphantom> <mi> y </mi> </mphantom> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> |
<mfenced>
The mfenced
element provides a convenient form
in which to express common constructs involving fences (i.e. braces,
brackets, and parentheses), possibly including separators (such as
comma) between the arguments.
For example, <mfenced> <mi>x</mi> </mfenced>
renders as "(x)" and is equivalent to
<mrow> <mo> ( </mo> <mi>x</mi> <mo> ) </mo> </mrow> |
and
<mfenced> <mi>x</mi> <mi>y</mi> </mfenced>
renders as "(x, y)"
and is equivalent to
<mrow> <mo> ( </mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo> ) </mo> </mrow> |
Individual fences or separators are represented using
mo
elements, as described in Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
. Thus, any mfenced
element is completely equivalent to an expanded form described below;
either form can be used in MathML, at the convenience of an author or
of a MathML-generating program. A MathML renderer is required to
render either of these forms in exactly the same way.
In general, an mfenced
element can contain
zero or more arguments, and will enclose them between fences in an
mrow
; if there is more than one argument, it will
insert separators between adjacent arguments, using an additional
nested mrow
around the arguments and separators
for proper grouping (Section 3.3.1 Horizontally Group Sub-Expressions
<mrow>
). The general expanded form is
shown below. The fences and separators will be parentheses and comma
by default, but can be changed using attributes, as shown in the
following table.
mfenced
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
The delimiters and separators should be drawn using the color specified by mathcolor
.
Name | values | default |
---|---|---|
open | string | ( |
Specifies the opening delimiter.
Since it is used as the content of an mo element, any whitespace
will be trimmed and collapsed as described in Section 2.1.7 Collapsing Whitespace in Input.
|
||
close | string | ) |
Specifies the closing delimiter.
Since it is used as the content of an mo element, any whitespace
will be trimmed and collapsed as described in Section 2.1.7 Collapsing Whitespace in Input.
|
||
separators | string | , |
Specifies a sequence of zero or more separator characters, optionally separated by whitespace.
Each pair of arguments is displayed separated by the corresponding separator
(none appears after the last argument).
If there are too many separators, the excess are ignored;
if there are too few, the last separator is repeated.
Any whitespace within separators is ignored.
|
A generic mfenced
element, with all attributes
explicit, looks as follows:
<mfenced open="opening-fence" close="closing-fence" separators="sep#1 sep#2 ... sep#(n-1)" > arg#1 ... arg#n </mfenced>
In an RTL directionality context, since the initial text
direction is RTL, characters in the open
and close
attributes that have a mirroring counterpart will be rendered in that
mirrored form. In particular, the default values will render correctly
as a parenthesized sequence in both LTR and RTL contexts.
The general mfenced
element shown above is
equivalent to the following expanded form:
<mrow> <mo fence="true"> opening-fence </mo> <mrow> arg#1 <mo separator="true"> sep#1 </mo> ... <mo separator="true"> sep#(n-1) </mo> arg#n </mrow> <mo fence="true"> closing-fence </mo> </mrow>
Each argument except the last is followed by a separator. The inner
mrow
is added for proper grouping, as described in
Section 3.3.1 Horizontally Group Sub-Expressions
<mrow>
.
When there is only one argument, the above form has no separators;
since <mrow> arg#1 </mrow>
is equivalent to
arg#1
(as described in Section 3.3.1 Horizontally Group Sub-Expressions
<mrow>
), this case is also equivalent to:
<mrow> <mo fence="true"> opening-fence </mo> arg#1 <mo fence="true"> closing-fence </mo> </mrow>
If there are too many separator characters, the extra ones are
ignored. If separator characters are given, but there are too few, the
last one is repeated as necessary. Thus, the default value of
separators
="," is equivalent to
separators
=",,", separators
=",,,", etc. If
there are no separator characters provided but some are needed, for
example if separators
=" " or "" and there is more than
one argument, then no separator elements are inserted at all — that
is, the elements <mo separator="true"> sep#i
</mo>
are left out entirely. Note that this is different
from inserting separators consisting of mo
elements with empty content.
Finally, for the case with no arguments, i.e.
<mfenced open="opening-fence" close="closing-fence" separators="anything" > </mfenced>
the equivalent expanded form is defined to include just
the fences within an mrow
:
<mrow> <mo fence="true"> opening-fence </mo> <mo fence="true"> closing-fence </mo> </mrow>
Note that not all "fenced expressions" can be encoded by an
mfenced
element. Such exceptional expressions
include those with an "embellished" separator or fence or one
enclosed in an mstyle
element, a missing or extra
separator or fence, or a separator with multiple content
characters. In these cases, it is necessary to encode the expression
using an appropriately modified version of an expanded form. As
discussed above, it is always permissible to use the expanded form
directly, even when it is not necessary. In particular, authors cannot
be guaranteed that MathML preprocessors won't replace occurrences of
mfenced
with equivalent expanded forms.
Note that the equivalent expanded forms shown above include
attributes on the mo
elements that identify them as fences or
separators. Since the most common choices of fences and separators
already occur in the operator dictionary with those attributes,
authors would not normally need to specify those attributes explicitly
when using the expanded form directly. Also, the rules for the default
form
attribute (Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
) cause the
opening and closing fences to be effectively given the values
form
="prefix"
and
form
="postfix"
respectively, and the
separators to be given the value
form
="infix"
.
Note that it would be incorrect to use mfenced
with a separator of, for instance, "+", as an abbreviation for an
expression using "+" as an ordinary operator, e.g.
<mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> </mrow> |
This is because the +
signs would be treated as separators,
not infix operators. That is, it would render as if they were marked up as
<mo separator="true">+</mo>
, which might therefore
render inappropriately.
(a+b)
<mfenced> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfenced> |
Note that the above mrow
is necessary so that
the mfenced
has just one argument. Without it, this
would render incorrectly as "(a, +,
b)".
[0,1)
<mfenced open="["> <mn> 0 </mn> <mn> 1 </mn> </mfenced> |
f(x,y)
<mrow> <mi> f </mi> <mo> ⁡<!--FUNCTION APPLICATION--> </mo> <mfenced> <mi> x </mi> <mi> y </mi> </mfenced> </mrow> |
<menclose>
The menclose
element renders its content
inside the enclosing notation specified by its notation
attribute.
menclose
accepts
a single argument possibly being an inferred mrow
of multiple children;
see Section 3.1.3 Required Arguments.
menclose
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
The notations should be drawn using the color specified by mathcolor
.
The values allowed for notation
are open-ended.
Conforming renderers may ignore any value they do not handle, although
renderers are encouraged to render as many of the values listed below as
possible.
Name | values | default |
---|---|---|
notation | ("longdiv" | "actuarial" | "phasorangle" | "radical" | "box" | "roundedbox" | "circle" | "left" | "right" | "top" | "bottom" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike" | "northeastarrow" | "madruwb" | text) + | longdiv |
Specifies a space separated list of notations to be used to enclose the children. See below for a description of each type of notation. |
Any number of values can be given for
notation
separated by whitespace; all of those given and
understood by a MathML renderer should be rendered.
Each should be rendered as if the others were not present; they should not nest one inside of the other. For example,
notation
="circle box" should
result in circle and a box around the contents of menclose
; the circle and box may overlap. This is shown in the first example below.
Of the predefined notations, only the following are
affected by the directionality (see Section 3.1.5.1 Overall Directionality of Mathematics Formulas):
"radical"
"phasorangle"
When notation
has the value "longdiv",
the contents are drawn enclosed by a long division symbol.
MathML 3 adds the mlongdiv
element
(Section 3.6.2 Long Division <mlongdiv>
). This element supports notations for long division
used in several countries and can be used to create a complete example of
long division as shown in Section 3.6.8.3 Long Division.
When notation
is specified as
"actuarial", the contents are drawn enclosed by an
actuarial symbol. A similar result can be achieved
with the value "top right". The case of
notation
="radical" is equivalent to the
msqrt
schema.
The values "box", "roundedbox", and "circle" should enclose the contents as indicated by the values. The amount of distance between the box, roundedbox, or circle, and the contents are not specified by MathML, and is left to the renderer. In practice, paddings on each side of 0.4em in the horizontal direction and .5ex in the vertical direction seem to work well.
The values "left",
"right", "top" and
"bottom" should result in lines drawn on those sides of
the contents. The values "northeastarrow",
"updiagonalstrike",
"downdiagonalstrike", "verticalstrike"
and "horizontalstrike" should result in the indicated
strikeout lines being superimposed over the content of the
menclose
, e.g. a strikeout that extends from the lower left
corner to the upper right corner of the menclose
element for
"updiagonalstrike", etc.
The value "northeastarrow" is a recommended value to implement because it can be
used to implement TeX's \cancelto command. If a renderer implements other arrows for menclose
, it is recommended that the arrow names are chosen from the following full set of names for consistancy and standardization
among renderers:
"uparrow"
"rightarrow"
"downarrow"
"leftarrow"
"northwestarrow"
"southwestarrow"
"southeastarrow"
"northeastarrow"
"updownarrow"
"leftrightarrow"
"northwestsoutheastarrow"
"northeastsouthwestarrow"
The value "madruwb" should generate an enclosure representing an Arabic factorial (‘madruwb’ is the transliteration of the Arabic مضروب for factorial). This is shown in the third example below.
The baseline of an menclose
element is the baseline of its child (which might be an implied mrow
).
An example of using multiple attributes is
<menclose notation='circle box'> <mi> x </mi><mo> + </mo><mi> y </mi> </menclose> |
which renders with the box and circle overlapping roughly as
.
An example of using menclose
for actuarial
notation is
<msub> <mi>a</mi> <mrow> <menclose notation='actuarial'> <mi>n</mi> </menclose> <mo>⁣<!--INVISIBLE SEPARATOR--></mo> <mi>i</mi> </mrow> </msub> |
which renders roughly as
An example of "phasorangle", which is used in circuit analysis, is:
<mi>C</mi> <mrow> <menclose notation='phasorangle'> <mrow> <mo>−<!--MINUS SIGN--></mo> <mfrac> <mi>π<!--GREEK SMALL LETTER PI--></mi> <mn>2</mn> </mfrac> </mrow> </menclose> </mrow> |
which renders roughly as
An example of "madruwb" is:
<menclose notation="madruwb"> <mn>12</mn> </menclose> |
which renders roughly as
.
The elements described in this section position one or more scripts around a base. Attaching various kinds of scripts and embellishments to symbols is a very common notational device in mathematics. For purely visual layout, a single general-purpose element could suffice for positioning scripts and embellishments in any of the traditional script locations around a given base. However, in order to capture the abstract structure of common notation better, MathML provides several more specialized scripting elements.
In addition to sub/superscript elements, MathML has overscript
and underscript elements that place scripts above and below the base. These
elements can be used to place limits on large operators, or for placing
accents and lines above or below the base. The rules for rendering accents
differ from those for overscripts and underscripts, and this difference can
be controlled with the accent
and accentunder
attributes, as described in the appropriate
sections below.
Rendering of scripts is affected by the scriptlevel
and displaystyle
attributes, which are part of the environment inherited by the rendering
process of every MathML expression, and are described in Section 3.1.6 Displaystyle and Scriptlevel.
These attributes cannot be given explicitly on a scripting element, but can be
specified on the start tag of a surrounding mstyle
element if desired.
MathML also provides an element for attachment of tensor indices. Tensor indices are distinct from ordinary subscripts and superscripts in that they must align in vertical columns. Tensor indices can also occur in prescript positions. Note that ordinary scripts follow the base (on the right in LTR context, but on the left in RTL context); prescripts precede the base (on the left (right) in LTR (RTL) context).
Because presentation elements should be used to describe the abstract notational structure of expressions, it is important that the base expression in all "scripting" elements (i.e. the first argument expression) should be the entire expression that is being scripted, not just the trailing character. For example, (x+y)2 should be written as:
<msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> |
<msub>
The msub
element attaches a subscript to a base using the syntax
<msub> base subscript </msub>
It increments scriptlevel
by 1, and sets displaystyle
to
"false", within subscript, but leaves both attributes
unchanged within base. (See Section 3.1.6 Displaystyle and Scriptlevel.)
msub
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
subscriptshift | length | automatic |
Specifies the minimum amount to shift the baseline of subscript down; the default is for the rendering agent to use its own positioning rules. |
<msup>
The msup
element attaches a superscript to a base using the syntax
<msup> base superscript </msup>
It increments scriptlevel
by 1, and sets displaystyle
to "false", within
superscript, but leaves both attributes unchanged within
base. (See Section 3.1.6 Displaystyle and Scriptlevel.)
msup
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
superscriptshift | length | automatic |
Specifies the minimum amount to shift the baseline of superscript up; the default is for the rendering agent to use its own positioning rules. |
<msubsup>
The msubsup
element is used to attach both a subscript and
superscript to a base expression.
<msubsup> base subscript superscript </msubsup>
It increments scriptlevel
by 1, and sets displaystyle
to
"false", within subscript and superscript,
but leaves both attributes unchanged within base.
(See Section 3.1.6 Displaystyle and Scriptlevel.)
Note that both scripts are positioned tight against the base as shown here
versus the staggered positioning of nested scripts as shown here
;
the latter can be achieved by nesting an
msub
inside an msup
.
msubsup
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
subscriptshift | length | automatic |
Specifies the minimum amount to shift the baseline of subscript down; the default is for the rendering agent to use its own positioning rules. | ||
superscriptshift | length | automatic |
Specifies the minimum amount to shift the baseline of superscript up; the default is for the rendering agent to use its own positioning rules. |
The msubsup
is most commonly used for adding
sub/superscript pairs to identifiers as illustrated above. However,
another important use is placing limits on certain large operators
whose limits are traditionally displayed in the script positions even
when rendered in display style. The most common of these is the
integral. For example,
would be represented as
<mrow> <msubsup> <mo> ∫<!--INTEGRAL--> </mo> <mn> 0 </mn> <mn> 1 </mn> </msubsup> <mrow> <msup> <mi> ⅇ<!--DOUBLE-STRUCK ITALIC SMALL E--> </mi> <mi> x </mi> </msup> <mo> ⁢<!--INVISIBLE TIMES--> </mo> <mrow> <mo> ⅆ<!--DOUBLE-STRUCK ITALIC SMALL D--> </mo> <mi> x </mi> </mrow> </mrow> </mrow> |
<munder>
The munder
element attaches an accent or limit placed under a base using the syntax
<munder> base underscript </munder>
It always sets displaystyle
to "false" within the underscript,
but increments scriptlevel
by 1 only when accentunder
is "false".
Within base, it always leaves both attributes unchanged.
(See Section 3.1.6 Displaystyle and Scriptlevel.)
If base is an operator with movablelimits
="true"
(or an embellished operator whose mo
element core has movablelimits
="true"),
and displaystyle
="false",
then underscript is drawn in a subscript position.
In this case, the accentunder
attribute is ignored.
This is often used for limits on symbols such as ∑
.
munder
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
accentunder | "true" | "false" | automatic |
Specifies whether underscript is drawn as an "accent" or as a limit.
An accent is drawn the same size as the base (without incrementing scriptlevel )
and is drawn closer to the base.
|
||
align | "left" | "right" | "center" | center |
Specifies whether the script is aligned left, center, or right under/over the base. As specfied in Section 3.2.5.8.3 Horizontal Stretching Rules, the core of underscripts that are embellished operators should stretch to cover the base, but the alignment is based on the entire underscript. |
The default value of accentunder
is false, unless
underscript is an mo
element or an
embellished operator (see Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
). If
underscript is an mo
element, the
value of its accent
attribute is used as the default
value of accentunder
. If underscript is an
embellished operator, the accent
attribute of the
mo
element at its core is used as the default
value. As with all attributes, an explicitly given value overrides
the default.
Here is an example (accent versus underscript):
versus
.
The MathML representation for this example is shown below.
The MathML representation for the example shown above is:
<mrow> <munder accentunder="true"> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ⏟<!--BOTTOM CURLY BRACKET--> </mo> </munder> <mtext> <!--NO-BREAK SPACE-->versus <!--NO-BREAK SPACE--></mtext> <munder accentunder="false"> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ⏟<!--BOTTOM CURLY BRACKET--> </mo> </munder> </mrow> |
<mover>
The mover
element attaches an accent or limit placed over a base using the syntax
<mover> base overscript </mover>
It always sets displaystyle
to "false" within overscript,
but increments scriptlevel
by 1 only when accent
is "false".
Within base, it always leaves both attributes unchanged.
(See Section 3.1.6 Displaystyle and Scriptlevel.)
If base is an operator with movablelimits
="true"
(or an embellished operator whose mo
element core has movablelimits
="true"),
and displaystyle
="false",
then overscript is drawn in a superscript position.
In this case, the accent
attribute is ignored.
This is often used for limits on symbols such as ∑
.
mover
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
accent | "true" | "false" | automatic |
Specifies whether overscript is drawn as an "accent" or as a limit.
An accent is drawn the same size as the base (without incrementing scriptlevel )
and is drawn closer to the base.
|
||
align | "left" | "right" | "center" | center |
Specifies whether the script is aligned left, center, or right under/over the base. As specfied in Section 3.2.5.8.3 Horizontal Stretching Rules, the core of overscripts that are embellished operators should stretch to cover the base, but the alignment is based on the entire overscript. |
The difference between an accent versus limit is shown here:
versus
.
These differences also apply to "mathematical accents" such as
bars or braces over expressions:
versus
.
The MathML representation for each of these examples is shown below.
The default value of accent is false, unless
overscript is an mo
element or an
embellished operator (see Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
). If
overscript is an mo
element, the value
of its accent
attribute is used as the default value
of accent
for mover
. If
overscript is an embellished operator, the accent
attribute of the mo
element at its core is used as the default value.
The MathML representation for the examples shown above is:
<mrow> <mover accent="true"> <mi> x </mi> <mo> ^<!--CIRCUMFLEX ACCENT--> </mo> </mover> <mtext> <!--NO-BREAK SPACE-->versus <!--NO-BREAK SPACE--></mtext> <mover accent="false"> <mi> x </mi> <mo> ^<!--CIRCUMFLEX ACCENT--> </mo> </mover> </mrow> |
<mrow> <mover accent="true"> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ⏞<!--TOP CURLY BRACKET--> </mo> </mover> <mtext> <!--NO-BREAK SPACE-->versus <!--NO-BREAK SPACE--></mtext> <mover accent="false"> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ⏞<!--TOP CURLY BRACKET--> </mo> </mover> </mrow> |
<munderover>
The munderover
element attaches accents or limits placed both over and under a base using the syntax
<munderover> base underscript overscript </munderover>
It always sets displaystyle
to "false"
within underscript and overscript,
but increments scriptlevel
by 1 only when
accentunder
or accent
, respectively, are "false".
Within base, it always leaves both attributes unchanged.
(see Section 3.1.6 Displaystyle and Scriptlevel).
If base is an operator with movablelimits
="true"
(or an embellished operator whose mo
element core has movablelimits
="true"),
and displaystyle
="false",
then underscript and overscript are drawn in a subscript and superscript position,
respectively. In this case, the accentunder
and accent
attributes are ignored.
This is often used for limits on symbols such as ∑
.
munderover
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
accent | "true" | "false" | automatic |
Specifies whether overscript is drawn as an "accent" or as a limit.
An accent is drawn the same size as the base (without incrementing scriptlevel )
and is drawn closer to the base.
|
||
accentunder | "true" | "false" | automatic |
Specifies whether underscript is drawn as an "accent" or as a limit.
An accent is drawn the same size as the base (without incrementing scriptlevel )
and is drawn closer to the base.
|
||
align | "left" | "right" | "center" | center |
Specifies whether the scripts are aligned left, center, or right under/over the base. As specfied in Section 3.2.5.8.3 Horizontal Stretching Rules, the core of underscripts and overscripts that are embellished operators should stretch to cover the base, but the alignment is based on the entire underscript or overscript. |
The munderover
element is used instead of separate
munder
and mover
elements so that the
underscript and overscript are vertically spaced equally in relation
to the base and so that they follow the slant of the base as in the
second expression shown below:
versus
.
The MathML representation for this example is shown below.
The difference in the vertical spacing is too small to be noticed on a low resolution display at a normal font size, but is noticeable on a higher resolution device such as a printer and when using large font sizes. In addition to the visual differences, attaching both the underscript and overscript to the same base more accurately reflects the semantics of the expression.
The defaults for accent
and accentunder
are computed in the same way as for
munder
and
mover
, respectively.
The MathML representation for the example shown above with the first
expression made using separate munder
and
mover
elements, and the second one using an
munderover
element, is:
<mrow> <mover> <munder> <mo> ∫<!--INTEGRAL--> </mo> <mn> 0 </mn> </munder> <mi> ∞<!--INFINITY--> </mi> </mover> <mtext> <!--NO-BREAK SPACE-->versus <!--NO-BREAK SPACE--></mtext> <munderover> <mo> ∫<!--INTEGRAL--> </mo> <mn> 0 </mn> <mi> ∞<!--INFINITY--> </mi> </munderover> </mrow> |
<mmultiscripts>
,
<mprescripts/>
,
<none/>
Presubscripts and tensor notations are represented by a single
element, mmultiscripts
, using the syntax:
<mmultiscripts> base (subscript superscript)* [ <mprescripts/> (presubscript presuperscript)* ] </mmultiscripts>
This element allows the representation of any number of vertically-aligned pairs of subscripts
and superscripts, attached to one base expression. It supports both
postscripts and
prescripts.
Missing scripts can be represented by the empty element
none
.
The prescripts are optional, and when present are given after the postscripts, because prescripts are relatively rare compared to tensor notation.
The argument sequence consists of the base followed by zero or more
pairs of vertically-aligned subscripts and superscripts (in that
order) that represent all of the postscripts. This list is optionally
followed by an empty element mprescripts
and a
list of zero or more pairs of vertically-aligned presubscripts and
presuperscripts that represent all of the prescripts. The pair lists
for postscripts and prescripts are displayed in the same order as the
directional context (ie. left-to-right order in LTR context). If
no subscript or superscript should be rendered in a given position,
then the empty element none
should be used in
that position.
The base, subscripts, superscripts, the optional separator element
mprescripts
, the presubscripts, and the
presuperscripts, are all direct sub-expressions of the
mmultiscripts
element, i.e. they are all at the
same level of the expression tree. Whether a script argument is a
subscript or a superscript, or whether it is a presubscript or a
presuperscript is determined by whether it occurs in an even-numbered
or odd-numbered argument position, respectively, ignoring the empty
element mprescripts
itself when determining the
position. The first argument, the base, is considered to be in
position 1. The total number of arguments must be odd, if
mprescripts
is not given, or even, if it is.
The empty element mprescripts
is only allowed as direct sub-expression
of mmultiscripts
.
Same as the attributes of msubsup
. See
Section 3.4.3.2 Attributes.
The mmultiscripts
element increments scriptlevel
by 1, and sets displaystyle
to "false", within
each of its arguments except base, but leaves both attributes
unchanged within base. (See Section 3.1.6 Displaystyle and Scriptlevel.)
Two examples of the use of mmultiscripts
are:
0F1(;a;z).
<mrow> <mmultiscripts> <mi> F </mi> <mn> 1 </mn> <none/> <mprescripts/> <mn> 0 </mn> <none/> </mmultiscripts> <mo> ⁡<!--FUNCTION APPLICATION--> </mo> <mrow> <mo> ( </mo> <mrow> <mo> ; </mo> <mi> a </mi> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> |
(where k and l are different indices)
<mmultiscripts> <mi> R </mi> <mi> i </mi> <none/> <none/> <mi> j </mi> <mi> k </mi> <none/> <mi> l </mi> <none/> </mmultiscripts> |
An additional example of mmultiscripts
shows how the binomial
coefficient
can be displayed in Arabic style
<mstyle dir="rtl"> <mmultiscripts><mo>ل<!--ARABIC LETTER LAM--></mo> <mn>12</mn><none/> <mprescripts/> <none/><mn>5</mn> </mmultiscripts> </mstyle> |
Matrices, arrays and other table-like mathematical notation are marked
up using mtable
,
mtr
, mlabeledtr
and
mtd
elements. These elements are similar to the
table
, tr
and td
elements of HTML, except that they provide
specialized attributes for the fine layout control
necessary for commutative diagrams, block matrices and so on.
While the two-dimensional layouts used for elementary math such as addition and multiplication
are somewhat similar to tables, they differ in important ways.
For layout and for accessibility reasons, the mstack
and mlongdiv
elements discussed
in Section 3.6 Elementary Math should be used for elementary math notations.
In addition to the table elements mentioned above, the mlabeledtr
element is used for labeling rows
of a table. This is useful for numbered equations.
The first child of mlabeledtr
is the label.
A label is somewhat special in that it is not considered an expression
in the matrix and is not counted when determining the number of columns
in that row.
<mtable>
A matrix or table is specified using the mtable
element. Inside of the mtable
element, only mtr
or mlabeledtr
elements may appear.
(In MathML 1.x, the mtable
was allowed to ‘infer’ mtr
elements around its arguments,
and the mtr
element could infer mtd
elements.
This behaviour is deprecated.)
Table rows that have fewer columns than other rows of the same
table (whether the other rows precede or follow them) are effectively
padded on the right (or left in RTL context) with empty mtd
elements so
that the number of columns in each row equals the maximum number of
columns in any row of the table. Note that the use of
mtd
elements with non-default values of the
rowspan
or columnspan
attributes may affect
the number of mtd
elements that should be given
in subsequent mtr
elements to cover a given
number of columns.
Note also that the label in an mlabeledtr
element
is not considered a column in the table.
MathML does not specify a table layout algorithm. In
particular, it is the responsibility of a MathML renderer to resolve
conflicts between the width
attribute and other
constraints on the width of a table, such as explicit values for columnwidth
attributes,
and minimum sizes for table cell contents. For a discussion of table layout algorithms, see
Cascading
Style Sheets, level 2.
mtable
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Any rules drawn as part of the mtable
should be drawn using the color
specified by mathcolor
.
Name | values | default |
---|---|---|
align | ("top" | "bottom" | "center" | "baseline" | "axis"), rownumber? | axis |
specifies the vertical alignment of the table with respect to its environment.
"axis" means to align the vertical center of the table on
the environment's axis.
(The axis of an equation is an alignment line used by typesetters.
It is the line on which a minus sign typically lies.)
"center" and "baseline" both mean to align the center of the table
on the environment's baseline.
"top" or "bottom" aligns the top or bottom of the table on the environment's baseline.
If the align attribute value ends with a rownumber,
the specified row (counting from 1 for the top row), rather than the table as a whole, is aligned in the way described above
with the exceptions noted below.
If rownumber is negative, it counts rows from the bottom.
When the value of rownumber is out of range or not an integer, it is ignored.
If a row number is specified and the alignment value is "baseline" or "axis",
the row's baseline or axis is used for alignment. Note this is only well defined when the rowalign
value is "baseline" or "axis"; MathML does not specify how
"baseline" or "axis" alignment should occur for other values of rowalign .
|
||
rowalign | ("top" | "bottom" | "center" | "baseline" | "axis") + | baseline |
specifies the vertical alignment of the cells with respect to other cells within the same row: "top" aligns the tops of each entry across the row; "bottom" aligns the bottoms of the cells, "center" centers the cells; "baseline" aligns the baselines of the cells; "axis" aligns the axis of each cells. (See the note below about multiple values). | ||
columnalign | ("left" | "center" | "right") + | center |
specifies the horizontal alignment of the cells with respect to other cells within the same column: "left" aligns the left side of the cells; "center" centers each cells; "right" aligns the right side of the cells. (See the note below about multiple values). | ||
groupalign | group-alignment-list-list | {left} |
[this attribute is described with the alignment elements, maligngroup and malignmark ,
in Section 3.5.5 Alignment Markers
<maligngroup/> , <malignmark/> .]
|
||
alignmentscope | ("true" | "false") + | true |
[this attribute is described with the alignment elements, maligngroup and malignmark ,
in Section 3.5.5 Alignment Markers
<maligngroup/> , <malignmark/> .]
|
||
columnwidth | ("auto" | length | "fit") + | auto |
specifies how wide a column should be:
"auto" means that the column should be as wide as needed;
an explicit length means that the column is exactly that wide and the contents of that column are made to fit
by linewrapping or clipping at the discretion of the renderer;
"fit" means that the page width
remaining after subtracting the "auto" or fixed width columns
is divided equally among the "fit" columns.
If insufficient room remains to hold the
contents of the "fit" columns, renderers may
linewrap or clip the contents of the "fit" columns.
Note that when the columnwidth is specified as
a percentage, the value is relative to the width of the table, not
as a percentage of the default (which is "auto"). That
is, a renderer should try to adjust the width of the column so that it
covers the specified percentage of the entire table width.
(See the note below about multiple values).
|
||
width | "auto" | length | auto |
specifies the desired width of the entire table and is intended for visual user agents. When the value is a percentage value or number without unit, the value is relative to the horizontal space that a MathML renderer has available , this is the current target width as used for linebreaking as specified in Section 3.1.7 Linebreaking of Expressions; this allows the author to specify, for example, a table being full width of the display. When the value is "auto", the MathML renderer should calculate the table width from its contents using whatever layout algorithm it chooses. | ||
rowspacing | (length) + | 1.0ex |
specifies how much space to add between rows. (See the note below about multiple values). | ||
columnspacing | (length) + | 0.8em |
specifies how much space to add between columns. (See the note below about multiple values). | ||
rowlines | ("none" | "solid" | "dashed") + | none |
specifies whether and what kind of lines should be added between each row: "none" means no lines; "solid" means solid lines; "dashed" means dashed lines (how the dashes are spaced is implementation dependent). (See the note below about multiple values). | ||
columnlines | ("none" | "solid" | "dashed") + | none |
specifies whether and what kind of lines should be added between each column: "none" means no lines; "solid" means solid lines; "dashed" means dashed lines (how the dashes are spaced is implementation dependent). (See the note below about multiple values). | ||
frame | "none" | "solid" | "dashed" | none |
specifies whether and what kind of lines should be drawn around the table. "none" means no lines; "solid" means solid lines; "dashed" means dashed lines (how the dashes are spaced is implementation dependent). | ||
framespacing | length, length | 0.4em 0.5ex |
specifies the additional spacing added between the table and frame,
if frame is not "none".
The first value specifies the spacing on the right and left;
the second value specifies the spacing above and below.
|
||
equalrows | "true" | "false" | false |
specifies whether to force all rows to have the same total height. | ||
equalcolumns | "true" | "false" | false |
specifies whether to force all columns to have the same total width. | ||
displaystyle | "true" | "false" | false |
specifies the value of displaystyle within each cell,
(scriptlevel is not changed);
see Section 3.1.6 Displaystyle and Scriptlevel.
|
||
side | "left" | "right" | "leftoverlap" | "rightoverlap" | right |
specifies on what side of the table labels from enclosed mlabeledtr (if any) should be placed.
The variants "leftoverlap" and "rightoverlap"
are useful when the table fits with the allowed width
when the labels are omitted, but not when they are included:
in such cases, the labels will overlap the row placed above it
if the rowalign for that row is "top",
otherwise it is placed below it.
|
||
minlabelspacing | length | 0.8em |
specifies the minimum space allowed between a label and the adjacent cell in the row. |
In the above specifications for attributes affecting rows
(respectively, columns, or the gaps between rows or columns),
the notation (...)+
means that multiple values can be given for the attribute
as a space separated list (see Section 2.1.5 MathML Attribute Values).
In this context, a single value specifies the value to be used for all rows (resp., columns or gaps).
A list of values are taken to apply to corresponding rows (resp., columns or gaps)
in order, that is starting from the top row for rows or first column (left or right,
depending on directionality) for columns.
If there are more rows (resp., columns or gaps) than supplied values, the last value is repeated as needed.
If there are too many values supplied, the excess are ignored.
Note that none of the areas occupied by lines
frame
, rowlines
and columnlines
,
nor the spacing framespacing
, rowspacing
or columnspacing
,
nor the label in mlabeledtr
are counted as rows or columns.
The displaystyle
attribute is allowed on the mtable
element to set the inherited value of the attribute. If the attribute is
not present, the mtable
element sets displaystyle
to
"false" within the table elements.
(See Section 3.1.6 Displaystyle and Scriptlevel.)
A 3 by 3 identity matrix could be represented as follows:
<mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> |
This might be rendered as:
Note that the parentheses must be represented explicitly; they are not
part of the mtable
element's rendering. This allows
use of other surrounding fences, such as brackets, or none at all.
<mtr>
An mtr
element represents one row in a table
or matrix. An mtr
element is only allowed as a
direct sub-expression of an mtable
element, and
specifies that its contents should form one row of the table. Each
argument of mtr
is placed in a different column
of the table, starting at the leftmost column in a LTR context or rightmost
column in a RTL context.
As described in Section 3.5.1 Table or Matrix
<mtable>
,
mtr
elements are
effectively padded with mtd
elements when they are shorter than other rows in a table.
mtr
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
rowalign | "top" | "bottom" | "center" | "baseline" | "axis" | inherited |
overrides, for this row, the vertical alignment of cells specified
by the rowalign attribute on the mtable .
|
||
columnalign | ("left" | "center" | "right") + | inherited |
overrides, for this row, the horizontal alignment of cells specified
by the columnalign attribute on the mtable .
|
||
groupalign | group-alignment-list-list | inherited |
[this attribute is described with the alignment elements, maligngroup and malignmark ,
in Section 3.5.5 Alignment Markers
<maligngroup/> , <malignmark/> .]
|
<mlabeledtr>
An mlabeledtr
element represents one row in
a table that has a label on either the left or right side, as
determined by the side
attribute. The label is
the first child of mlabeledtr
, and should be enclosed in an mtd
.
The rest of the children represent the contents of the row and are treated identically
to the children of mtr
; consequently all of the children
must be mtd
elements.
An mlabeledtr
element is only allowed as a
direct sub-expression of an mtable
element.
Each argument of mlabeledtr
except for the first
argument (the label) is placed in a different column
of the table, starting at the leftmost column.
Note that the label element is not considered to be a cell in the
table row. In particular, the label element is not taken into
consideration in the table layout for purposes of width and alignment
calculations. For example, in the case of an mlabeledtr
with a label and a single centered mtd
child, the child is first centered in the
enclosing mtable
, and then the label is
placed. Specifically, the child is not centered in the
space that remains in the table after placing the label.
While MathML does not specify an algorithm for placing labels,
implementers of visual renderers may find the following formatting
model useful. To place a label, an implementor might think in terms
of creating a larger table, with an extra column on both ends. The
columnwidth
attributes of both these border
columns would be set to "fit" so that they expand
to fill whatever space remains after the inner columns have been laid
out. Finally, depending on the values of side
and minlabelspacing
, the label is placed
in whatever border column is appropriate, possibly shifted down if
necessary, and aligned according to columnalignment
.
The attributes for mlabeledtr
are the same
as for mtr
. Unlike the attributes for the
mtable
element, attributes of
mlabeledtr
that apply to column elements
also apply to the label. For example, in a one column table,
<mlabeledtr rowalign='top'>
means that the label and other entries in the row are vertically aligned
along their top. To force a particular alignment on the label,
the appropriate attribute would normally be set on the
mtd
element that surrounds the label content.
One of the important uses of mlabeledtr
is
for numbered equations. In a mlabeledtr
, the
label represents the equation number and the elements in the row are
the equation being numbered. The side
and minlabelspacing
attributes of mtable
determine the placement of the equation
number.
In larger documents with many numbered equations, automatic
numbering becomes important. While automatic equation numbering and
automatically resolving references to equation numbers is outside the
scope of MathML, these problems can be addressed by the use of style
sheets or other means. The mlabeledtr construction provides support
for both of these functions in a way that is intended to facilitate
XSLT processing. The mlabeledtr
element can be
used to indicate the presence of a numbered equation, and the first
child can be changed to the current equation number, along with
incrementing the global equation number. For cross references, an
id
on either the mlabeledtr element or on the first element
itself could be used as a target of any link.
Alternatively, in a CSS context, one could use an empty mtd
as the first child of mlabeledtr
and use CSS counters and generated content
to fill in the equation number using a CSS style such as
body {counter-reset: eqnum;} mtd.eqnum {counter-increment: eqnum;} mtd.eqnum:before {content: "(" counter(eqnum) ")"}
<mtable> <mlabeledtr id='e-is-m-c-square'> <mtd> <mtext> (2.1) </mtext> </mtd> <mtd> <mrow> <mi>E</mi> <mo>=</mo> <mrow> <mi>m</mi> <mo>⁢<!--INVISIBLE TIMES--></mo> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </mrow> </mtd> </mlabeledtr> </mtable> |
This should be rendered as:
E = mc2 | (2.1) |
<mtd>
An mtd
element represents one entry, or cell, in a
table or matrix. An mtd
element is only
allowed as a direct sub-expression of an mtr
or an mlabeledtr
element.
The mtd
element accepts
a single argument possibly being an inferred mrow
of multiple children;
see Section 3.1.3 Required Arguments.
mtd
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
rowspan | positive-integer | 1 |
causes the cell to be treated as if it occupied the number of rows specified.
The corresponding mtd in the following "rowspan"-1 rows must be omitted.
The interpretation corresponds with the similar attributes for HTML 4.01 tables.
|
||
columnspan | positive-integer | 1 |
causes the cell to be treated as if it occupied the number of columns specified.
The following "rowspan"-1 mtd s must be omitted.
The interpretation corresponds with the similar attributes for HTML 4.01 tables.
|
||
rowalign | "top" | "bottom" | "center" | "baseline" | "axis" | inherited |
specifies the vertical alignment of this cell, overriding any value
specified on the containing mrow and mtable .
See the rowalign attribute of mtable .
|
||
columnalign | "left" | "center" | "right" | inherited |
specifies the horizontal alignment of this cell, overriding any value
specified on the containing mrow and mtable .
See the columnalign attribute of mtable .
|
||
groupalign | group-alignment-list | inherited |
[this attribute is described with the alignment elements, maligngroup and malignmark ,
in Section 3.5.5 Alignment Markers
<maligngroup/> , <malignmark/> .]
|
The rowspan
and columnspan
attributes
can be used around an mtd
element that represents
the label in a mlabeledtr
element.
Also, the label of a mlabeledtr
element is not
considered to be part of a previous rowspan
and
columnspan
.
<maligngroup/>
, <malignmark/>
Alignment markers are space-like elements (see Section 3.2.7 Space <mspace/>
) that can be used
to vertically align specified points within a column of MathML
expressions by the automatic insertion of the necessary amount of
horizontal space between specified sub-expressions.
The discussion that follows will use the example of a set of simultaneous equations that should be rendered with vertical alignment of the coefficients and variables of each term, by inserting spacing somewhat like that shown here:
8.44x + 55 y = 0 3.1 x - 0.7y = -1.1
If the example expressions shown above were arranged in a column but not aligned, they would appear as:
8.44x + 55y = 0 3.1x - 0.7y = -1.1
For audio renderers, it is suggested that the alignment elements produce the analogous behavior of altering the rhythm of pronunciation so that it is the same for several sub-expressions in a column, by the insertion of the appropriate time delays in place of the extra horizontal spacing described here.
The expressions whose parts are to be aligned (each equation, in the
example above) must be given as the table elements (i.e. as the mtd
elements) of one column of an
mtable
. To avoid confusion, the term "table
cell" rather than "table element" will be used in the
remainder of this section.
All interactions between alignment elements are limited to the
mtable
column they arise in. That is, every
column of a table specified by an mtable
element
acts as an "alignment scope" that contains within it all alignment
effects arising from its contents. It also excludes any interaction
between its own alignment elements and the alignment elements inside
any nested alignment scopes it might contain.
The reason mtable
columns are used as
alignment scopes is that they are the only general way in MathML to
arrange expressions into vertical columns. Future versions of MathML
may provide an malignscope
element that allows
an alignment scope to be created around any MathML element, but even
then, table columns would still sometimes need to act as alignment
scopes, and since they are not elements themselves, but rather are
made from corresponding parts of the content of several
mtr
elements, they could not individually be the
content of an alignment scope element.
An mtable
element can be given the attribute
alignmentscope
="false" to cause
its columns not to act as alignment scopes. This is discussed further at
the end of this section. Otherwise, the discussion in this section assumes
that this attribute has its default value of "true".
To cause alignment, it is necessary to specify, within each
expression to be aligned, the points to be aligned with corresponding
points in other expressions, and the beginning of each alignment
group of sub-expressions that can be horizontally shifted as a
unit to effect the alignment. Each alignment group must contain one
alignment point. It is also necessary to specify which expressions in
the column have no alignment groups at all, but are affected only by
the ordinary column alignment for that column of the table, i.e. by
the columnalign
attribute, described elsewhere.
The alignment groups start at the locations of invisible
maligngroup
elements, which are rendered with
zero width when they occur outside of an alignment scope, but within
an alignment scope are rendered with just enough horizontal space to
cause the desired alignment of the alignment group that follows
them. A simple algorithm by which a MathML application can achieve this is given
later. In the example above, each equation would have one
maligngroup
element before each coefficient,
variable, and operator on the left-hand side, one before the
=
sign, and one before the constant on the right-hand
side.
In general, a table cell containing n
maligngroup
elements contains n
alignment groups, with the ith group consisting of the
elements entirely after the ith
maligngroup
element and before the
(i+1)-th; no element within the table cell's content
should occur entirely before its first
maligngroup
element.
Note that the division into alignment groups does not
necessarily fit the nested expression structure of the MathML
expression containing the groups — that is, it is permissible for one
alignment group to consist of the end of one
mrow
, all of another one, and the beginning of a
third one, for example. This can be seen in the MathML markup for the
present example, given at the end of this section.
The nested expression structure formed by mrow
s
and other layout schemata should reflect the mathematical structure of the
expression, not the alignment-group structure, to make possible optimal
renderings and better automatic interpretations; see the discussion of
proper grouping in section Section 3.3.1 Horizontally Group Sub-Expressions
<mrow>
. Insertion of
alignment elements (or other space-like elements) should not alter the
correspondence between the structure of a MathML expression and the
structure of the mathematical expression it represents.
Although alignment groups need not
coincide with the nested expression structure of layout schemata,
there are nonetheless restrictions on where an maligngroup
element is allowed within a table cell. The maligngroup
element may only be contained within elements (directly or indirectly) of the following types
(which are themselves contained in the table cell):
an mrow
element, including an inferred
mrow
such as the one formed by a multi-child
mtd
element, but excluding mrow
which
contains a change of direction using the dir
attribute;
an mstyle
element
, but excluding those which change direction
using the dir
attribute;
an mphantom
element;
an mfenced
element;
an maction
element, though only its
selected sub-expression is checked;
a semantics
element.
These restrictions are intended to ensure that alignment can be unambiguously specified, while avoiding complexities involving things like overscripts, radical signs and fraction bars. They also ensure that a simple algorithm suffices to accomplish the desired alignment.
Note that some positions for an maligngroup
element, although legal, are not useful, such as an
maligngroup
element that is an argument of an
mfenced
element. Similarly, when inserting an maligngroup
element in an element whose arguments have positional significance, it
is necessary to introduce a new mrow
element
containing just the maligngroup
element and the child element
it precedes in order to preserve the proper expression structure. For
example, to insert an maligngroup
before the denominator
child of an mfrac
element, it is necessary to enclose the
maligngroup
and the denominator in an mrow
to
avoid introducing an illegal third child in the
mfrac
. In general, this will be necessary except
when the maligngroup
element is inserted directly into an
mrow
or into an element that can form an inferred
mrow
from its contents. See the warning about the legal
grouping of "space-like elements" in Section 3.2.7 Space <mspace/>
for an analogous example
involving malignmark
.
For the table cells that are divided into alignment groups, every
element in their content must be part of exactly one alignment group,
except for the elements from the above list that contain
maligngroup
elements inside them and the
maligngroup
elements themselves. This means
that, within any table cell containing alignment groups, the first
complete element must be an maligngroup
element,
though this may be preceded by the start tags of other elements.
This requirement removes a potential confusion about how to align
elements before the first maligngroup
element,
and makes it easy to identify table cells that are left out of their
column's alignment process entirely.
Note that it is not required that the table cells in a column that are divided into alignment groups each contain the same number of groups. If they don't, zero-width alignment groups are effectively added on the right side (or left side, in a RTL context) of each table cell that has fewer groups than other table cells in the same column.
Expressions in a column that are to have no alignment groups
should contain no maligngroup
elements. Expressions with no alignment groups are aligned using only
the columnalign
attribute that applies to the table
column as a whole, and are not affected by the groupalign
attribute described below. If such an expression is wider than the
column width needed for the table cells containing alignment groups,
all the table cells containing alignment groups will be shifted as a
unit within the column as described by the columnalign
attribute for that column. For example, a column heading with no
internal alignment could be added to the column of two equations given
above by preceding them with another table row containing an
mtext
element for the heading, and using the
default columnalign
="center" for the table, to
produce:
equations with aligned variables 8.44x + 55 y = 0 3.1 x - 0.7y = -1.1
or, with a shorter heading,
some equations 8.44x + 55 y = 0 3.1 x - 0.7y = -1.1
<malignmark/>
Each alignment group's alignment point can either be specified by
an malignmark
element anywhere within the
alignment group (except within another alignment scope wholly
contained inside it), or it is determined automatically from the
groupalign
attribute. The groupalign
attribute can be specified on the group's preceding
maligngroup
element or on its surrounding
mtd
, mtr
, or
mtable
elements. In typical cases, using the
groupalign
attribute is sufficient to describe the
desired alignment points, so no malignmark
elements need to be provided.
The malignmark
element indicates that the
alignment point should occur on the right edge of the preceding
element, or the left edge of the following element or character,
depending on the edge
attribute of
malignmark
. Note that it may be necessary to
introduce an mrow
to group an
malignmark
element with a neighboring element,
in order not to alter the argument count of the containing
element. (See the warning about the legal grouping of "space-like
elements" in Section 3.2.7 Space <mspace/>
).
When an malignmark
element is provided within an
alignment group, it can occur in an arbitrarily deeply nested element
within the group, as long as it is not within a nested alignment scope. It
is not subject to the same restrictions on location as maligngroup
elements. However, its immediate
surroundings need to be such that the element to its immediate right or
left (depending on its edge
attribute) can be
unambiguously identified. If no such element is present, renderers should
behave as if a zero-width element had been inserted there.
For the purposes of alignment, an element X is considered to be to the
immediate left of an element Y, and Y to the immediate right of X, whenever
X and Y are successive arguments of one (possibly inferred) mrow
element,
with X coming before Y (in a LTR context; with X coming after Y in a RTL context). In the case of
mfenced
elements, MathML applications should evaluate this
relation as if the mfenced
element had been
replaced by the equivalent expanded form involving mrow
. Similarly, an maction
element should be treated as if it were replaced by its currently selected
sub-expression. In all other cases, no relation of "to the immediate
left or right" is defined for two elements X and Y. However, in the
case of content elements interspersed in presentation markup, MathML applications
should attempt to evaluate this relation in a sensible way. For example, if
a renderer maintains an internal presentation structure for rendering
content elements, the relation could be evaluated with respect to
that. (See Chapter 4 Content Markup and Chapter 5 Mixing Markup Languages for Mathematical Expressions for further
details about mixing presentation and content markup.)
malignmark
elements are allowed to occur within
the content of token elements, such as mn
,
mi
, or mtext
. When this
occurs, the character immediately before or after the
malignmark
element will carry the alignment
point; in all other cases, the element to its immediate left or right
will carry the alignment point. The rationale for this is that it is
sometimes desirable to align on the edges of specific characters
within multi-character token elements.
If there is more than one malignmark
element
in an alignment group, all but the first one will be ignored. MathML
applications may wish to provide a mode in which they will warn about
this situation, but it is not an error, and should trigger no warnings
by default. The rationale for this is that it would
be inconvenient to have to remove all
unnecessary malignmark
elements from
automatically generated data, in certain cases, such as when they are
used to specify alignment on "decimal points" other than the '.'
character.
<malignmark/>
Attributes
malignmark
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements
(however, neither mathcolor
nor mathbackground
have any effect).
Name | values | default |
---|---|---|
edge | "left" | "right" | left |
see the discussion below. |
The
edge
attribute specifies whether the alignment point will be
found on the left or right edge of some element or character. The
precise location meant by "left edge" or "right edge" is discussed
below. If edge
="right", the alignment point is the right
edge of the element or character to the immediate left of the
malignmark
element. If edge
="left",
the alignment point is the left edge of the element or character to
the immediate right of the malignmark
element. Note that the attribute refers to the choice of edge rather
than to the direction in which to look for the element whose edge will
be used.
For malignmark
elements that occur within
the content of MathML token elements, the preceding or following
character in the token element's content is used; if there is no such
character, a zero-width character is effectively inserted for the
purpose of carrying the alignment point on its edge. For all other
malignmark
elements, the preceding or following
element is used; if there is no such element, a zero-width element is
effectively inserted to carry the alignment point.
The precise definition of the "left edge" or "right edge" of a character or glyph (e.g. whether it should coincide with an edge of the character's bounding box) is not specified by MathML, but is at the discretion of the renderer; the renderer is allowed to let the edge position depend on the character's context as well as on the character itself.
For proper alignment of columns of numbers (using groupalign
values of "left", "right", or "decimalpoint"), it is
likely to be desirable for the effective width (i.e. the distance between
the left and right edges) of decimal digits to be constant, even if their
bounding box widths are not constant (e.g. if "1" is narrower
than other digits). For other characters, such as letters and operators, it
may be desirable for the aligned edges to coincide with the bounding
box.
The "left edge" of a MathML element or alignment group
refers to the left edge of the leftmost glyph drawn to render the element
or group, except that explicit space represented by mspace
or mtext
elements
should also count as "glyphs" in this context, as should
glyphs that would be drawn if not for mphantom
elements around them. The "right edge" of an element or
alignment group is defined similarly.
<maligngroup/>
Attributes
maligngroup
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements
(however, neither mathcolor
nor mathbackground
have any effect).
Name | values | default |
---|---|---|
groupalign | "left" | "center" | "right" | "decimalpoint" | inherited |
see the discussion below. |
maligngroup
has one attribute,
groupalign
, which is used to determine the position of
its group's alignment point when no malignmark
element is present. The following discussion assumes that no
malignmark
element is found within a group.
In the example given at the beginning of this section, there is one
column of 2 table cells, with 7 alignment groups in each table cell;
thus there are 7 columns of alignment groups, with 2 groups, one above
the other, in each column. These columns of alignment groups should be
given the 7 groupalign
values "decimalpoint left left
decimalpoint left left decimalpoint", in that order. How to specify
this list of values for a table cell or table column as a whole, using
attributes on elements surrounding the
maligngroup
element is described later.
If groupalign
is "left",
"right", or "center", the alignment point is
defined to be at the group's left edge, at its right edge, or halfway
between these edges, respectively. The meanings of "left edge"
and "right edge" are as discussed above in relation to malignmark
.
If groupalign
is "decimalpoint",
the alignment point is the right edge of the character immediately before the
left-most 'decimal point', i.e. matching the character specified by
the decimalpoint
attribute of mstyle
(default ".", U+002E)
in the first mn
element found along
the alignment group's baseline. More precisely, the alignment group is
scanned recursively, depth-first, for the first mn
element, descending into all arguments of each element of the types
mrow
(including inferred
mrow
s), mstyle
,
mpadded
, mphantom
, menclose
,
mfenced
, or msqrt
,
descending into only the first argument of each "scripting" element
(msub
, msup
,
msubsup
, munder
,
mover
, munderover
,
mmultiscripts
) or of each
mroot
or semantics
element,
descending into only the selected sub-expression of each
maction
element, and skipping the content of all
other elements. The first mn
so found always
contains the alignment point, which is the right edge of the last
character before the first decimal point in the content of the
mn
element. If there is no decimal point in the
mn
element, the alignment point is the right edge
of the last character in the content. If the decimal point is the
first character of the mn
element's content, the
right edge of a zero-width character inserted before the decimal point
is used. If no mn
element is found, the right
edge of the entire alignment group is used (as for
groupalign
="right").
In order to permit alignment on decimal points in
cn
elements, a MathML application can convert a
content expression into a presentation expression that renders the
same way before searching for decimal points as described above.
Characters other than "." can be used as
"decimal points" for alignment by using mstyle
;
more arbitrary alignment points can chosen by embedding malignmark
elements
within the mn
token's content itself.
For any of the groupalign
values, if an explicit
malignmark
element is present anywhere within
the group, the position it specifies (described earlier) overrides the
automatic determination of alignment point from the
groupalign
value.
groupalign
values
It is not usually necessary to put a groupalign
attribute on every maligngroup
element. Since
this attribute is usually the same for every group in a column of
alignment groups to be aligned, it can be inherited from an attribute
on the mtable
that was used to set up the
alignment scope as a whole, or from the mtr
or
mtd
elements surrounding the alignment group. It
is inherited via an "inheritance path" that proceeds from
mtable
through successively contained
mtr
, mtd
, and
maligngroup
elements. There is exactly one
element of each of these kinds in this path from an
mtable
to any alignment group inside it. In
general, the value of groupalign
will be
inherited by any given alignment group from the innermost element
that surrounds the alignment group and provides an explicit
setting for this attribute. For example, if an
mtable
element specifies values for groupalign
and
a maligngroup
element within the table also specifies an
explicit groupalign
value, then then the value from the
maligngroup
takes priority.
Note, however, that each mtd
element needs, in
general, a list of groupalign
values, one for each
maligngroup
element inside it (from left to right, in an LTR context,
or from right to left in an RTL context), rather than just
a single value. Furthermore, an mtr
or
mtable
element needs, in general, a list of lists
of groupalign
values, since it spans multiple
mtable
columns, each potentially acting as an
alignment scope. Such lists of group-alignment values are specified
using the following syntax rules:
group-alignment = "left" | "right" | "center" | "decimalpoint" group-alignment-list = group-alignment + group-alignment-list-list = ( "{" group-alignment-list "}" ) +
As described in Section 2.1.5 MathML Attribute Values, |
separates
alternatives; +
represents optional repetition (i.e. 1 or
more copies of what precedes it), with extra values ignored and the
last value repeated if necessary to cover additional table columns or
alignment group columns; '{'
and '}'
represent literal braces; and (
and )
are
used for grouping, but do not literally appear in the attribute
value.
The permissible values of the groupalign
attribute of the
elements that have this attribute are specified using the above
syntax definitions as follows:
Element type | groupalign attribute syntax | default value |
---|---|---|
mtable |
group-alignment-list-list | {left} |
mtr |
group-alignment-list-list | inherited from mtable attribute |
mlabeledtr |
group-alignment-list-list | inherited from mtable attribute |
mtd |
group-alignment-list | inherited from within mtr attribute |
maligngroup |
group-alignment | inherited from within mtd attribute |
In the example near the beginning of this section, the group
alignment values could be specified on every mtd
element using groupalign
= "decimalpoint left left
decimalpoint left left decimalpoint", or on every
mtr
element using groupalign
=
"{decimalpoint left left decimalpoint left left decimalpoint}", or
(most conveniently) on the mtable
as a whole
using groupalign
= "{decimalpoint left left decimalpoint
left left decimalpoint}", which provides a single braced list of
group-alignment values for the single column of expressions to be
aligned.
The above rules are sufficient to explain the MathML representation of the example given near the start of this section. To repeat the example, the desired rendering is:
8.44x + 55 y = 0 3.1 x - 0.7y = -1.1
One way to represent that in MathML is:
<mtable groupalign= "{decimalpoint left left decimalpoint left left decimalpoint}"> <mtr> <mtd> <mrow> <mrow> <mrow> <maligngroup/> <mn> 8.44 </mn> <mo> ⁢<!--INVISIBLE TIMES--> </mo> <maligngroup/> <mi> x </mi> </mrow> <maligngroup/> <mo> + </mo> <mrow> <maligngroup/> <mn> 55 </mn> <mo> ⁢<!--INVISIBLE TIMES--> </mo> <maligngroup/> <mi> y </mi> </mrow> </mrow> <maligngroup/> <mo> = </mo> <maligngroup/> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mrow> <maligngroup/> <mn> 3.1 </mn> <mo> ⁢<!--INVISIBLE TIMES--> </mo> <maligngroup/> <mi> x </mi> </mrow> <maligngroup/> <mo> - </mo> <mrow> <maligngroup/> <mn> 0.7 </mn> <mo> ⁢<!--INVISIBLE TIMES--> </mo> <maligngroup/> <mi> y </mi> </mrow> </mrow> <maligngroup/> <mo> = </mo> <maligngroup/> <mrow> <mo> - </mo> <mn> 1.1 </mn> </mrow> </mrow> </mtd> </mtr> </mtable> |
The alignment elements maligngroup
and
malignmark
can occur outside of alignment
scopes, where they are ignored. The rationale behind this is that in
situations in which MathML is generated, or copied from another
document, without knowing whether it will be placed inside an
alignment scope, it would be inconvenient for this to be an error.
An mtable
element can be given the attribute alignmentscope
="false" to cause its
columns not to act as alignment scopes. In general, this attribute has the
syntax ("true" | "false") +
; if its value is a list of Boolean
values, each Boolean value applies to one column, with the last value
repeated if necessary to cover additional columns, or with extra values
ignored. Columns that are not alignment scopes are part of the alignment
scope surrounding the mtable
element, if there is
one. Use of alignmentscope
="false" allows nested tables to contain malignmark
elements for aligning the inner table in the
surrounding alignment scope.
As discussed above, processing of alignment for content elements is not well-defined, since MathML does not specify how content elements should be rendered. However, many MathML applications are likely to find it convenient to internally convert content elements to presentation elements that render the same way. Thus, as a general rule, even if a renderer does not perform such conversions internally, it is recommended that the alignment elements should be processed as if it did perform them.
A particularly important case for renderers to handle gracefully is the
interaction of alignment elements with the matrix
content element, since this element may or may not be internally converted
to an expression containing an mtable
element for
rendering. To partially resolve this ambiguity, it is suggested, but not
required, that if the matrix
element is converted
to an expression involving an mtable
element, that
the mtable
element be given the attribute alignmentscope
="false", which will
make the interaction of the matrix
element with the
alignment elements no different than that of a generic presentation element
(in particular, it will allow it to contain malignmark
elements that operate within the alignment
scopes created by the columns of an mtable
that
contains the matrix
element in one of its table
cells).
The effect of alignment elements within table cells that have
non-default values of the columnspan
or rowspan
attributes is not specified, except that such
use of alignment elements is not an error. Future versions of MathML may
specify the behavior of alignment elements in such table cells.
The effect of possible linebreaking of an mtable
element on the alignment elements is not specified.
A simple algorithm by which a MathML renderer can perform the
alignment specified in this section is given here. Since the alignment
specification is deterministic (except for the definition of the left
and right edges of a character), any correct MathML alignment
algorithm will have the same behavior as this one. Each
mtable
column (alignment scope) can be treated
independently; the algorithm given here applies to one
mtable
column, and takes into account the
alignment elements, the groupalign
attribute described in
this section, and the columnalign
attribute described
under mtable
(Section 3.5.1 Table or Matrix
<mtable>
).
First, a rendering is computed for the contents of each table cell
in the column, using zero width for all
maligngroup
and malignmark
elements. The final rendering will be identical except for horizontal
shifts applied to each alignment group and/or table cell. The
positions of alignment points specified by any
malignmark
elements are noted, and the remaining
alignment points are determined using groupalign
values.
For each alignment group, the horizontal positions of the left edge, alignment point, and right edge are noted, allowing the width of the group on each side of the alignment point (left and right) to be determined. The sum of these two "side-widths", i.e. the sum of the widths to the left and right of the alignment point, will equal the width of the alignment group.
Second, each column of alignment groups is scanned. The ith scan covers the ith alignment group in each table cell containing any alignment groups. Table cells with no alignment groups, or with fewer than i alignment groups, are ignored. Each scan computes two maximums over the alignment groups scanned: the maximum width to the left of the alignment point, and the maximum width to the right of the alignment point, of any alignment group scanned.
The sum of all the maximum widths computed (two for each column of alignment groups) gives one total width, which will be the width of each table cell containing alignment groups. Call the maximum number of alignment groups in one cell n; each such cell is divided into 2n horizontally adjacent sections, called L(i) and R(i) for i from 1 to n, using the 2n maximum side-widths computed above; for each i, the width of all sections called L(i) is the maximum width of any cell's ith alignment group to the left of its alignment point, and the width of all sections called R(i) is the maximum width of any cell's ith alignment group to the right of its alignment point.
Each alignment group is then shifted horizontally as a block to a unique position that places: in the section called L(i) that part of the ith group to the left of its alignment point; in the section called R(i) that part of the ith group to the right of its alignment point. This results in the alignment point of each ith group being on the boundary between adjacent sections L(i) and R(i), so that all alignment points of ith groups have the same horizontal position.
The widths of the table cells that contain no alignment groups were computed as part of the initial rendering, and may be different for each cell, and different from the single width used for cells containing alignment groups. The maximum of all the cell widths (for both kinds of cells) gives the width of the table column as a whole.
The position of each cell in the column is determined by the
applicable part of the value of the columnalign
attribute
of the innermost surrounding mtable
,
mtr
, or mtd
element that
has an explicit value for it, as described in the sections on those
elements. This may mean that the cells containing alignment groups
will be shifted within their column, in addition to their alignment
groups having been shifted within the cells as described above, but
since each such cell has the same width, it will be shifted the same
amount within the column, thus maintaining the vertical alignment of
the alignment points of the corresponding alignment groups in each
cell.
Mathematics used in the lower grades such as two-dimensional addition, multiplication, and long division tends to be tabular in nature. However, the specific notations used varies among countries much more than for higher level math. Furthermore, elementary math often presents examples in some intermediate state and MathML must be able to capture these intermediate or intentionally missing partial forms. Indeed, these constructs represent memory aids or procedural guides, as much as they represent ‘mathematics’.
The elements used for basic alignments in elementary math are:
mstack
msgroup
msrow
msline
mscarries
mscarry
mlongdiv
mstack
and mlongdiv
are the parent elements for all elementary
math layout.
Any children of mstack
, mlongdiv
, and msgroup
,
besides msrow
, msgroup
, mscarries
and msline
,
are treated as if implicitly surrounded by an msrow
(See Section 3.6.4 Rows in Elementary Math <msrow>
for more details about rows).
Since the primary use of these stacking constructs is to
stack rows of numbers aligned on their digits,
and since numbers are always formatted left-to-right,
the columns of an mstack are always processed left-to-right;
the overall directionality in effect (ie. the dir
attribute)
does not affect to the ordering of display of columns or carries in rows
and, in particular, does not affect the ordering of any operators within a row
(See Section 3.1.5 Directionality).
These elements are described in this section followed by examples of their use. In addition to two-dimensional addition, subtraction, multiplication, and long division, these elements can be used to represent several notations used for repeating decimals.
A very simple example of two-dimensional addition is shown below:
The MathML for this is:
<mstack> <mn>424</mn> <msrow> <mo>+</mo> <mn>33</mn> </msrow> <msline/> </mstack> |
Many more examples are given in Section 3.6.8 Elementary Math Examples.
<mstack>
mstack
is used to lay out rows of numbers that are aligned on each digit.
This is common in many elementary math notations such as 2D addition, subtraction, and multiplication.
The children of an mstack
represent rows, or groups of them,
to be stacked each below the previous row; there can be any number of rows.
An msrow
represents a row;
an msgroup
groups a set of rows together
so that their horizontal alignment can be adjusted together;
an mscarries
represents a set of carries to be
applied to the following row;
an msline
represents a line separating rows.
Any other element is treated as if implicitly surrounded by msrow
.
Each row contains ‘digits’ that are placed into columns.
(see Section 3.6.4 Rows in Elementary Math <msrow>
for further details).
The stackalign
attribute together with
the position
and shift
attributes of msgroup
,
mscarries
, and msrow
determine
to which column a character belongs.
The width of a column is the maximum of the widths of each ‘digit’ in that
column — carries do not participate in the
width calculation; they are treated as having zero width.
If an element is too wide to fit into a column, it overflows into the adjacent
column(s) as determined by the charalign
attribute.
If there is no character in a column, its width is taken to be the width of a 0
in the current language (in many fonts, all digits have the same width).
The method for laying out an mstack is:
The ‘digits’ in a row are determined.
All of the digits in a row are initially aligned according to the stackalign
value.
Each row is positioned relative to that alignment based on the position
attribute (if any)
that controls that row.
The maximum width of the digits in a column are determined and
shorter and wider entries in that column are aligned according to
the charalign
attribute.
The width and height of the mstack element are computed based on the rows and columns. Any overflow from a column is not used as part of that computation.
The baseline of the mstack element is determined by the align
attribute.
mstack
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
align | ("top" | "bottom" | "center" | "baseline" | "axis"), rownumber? | baseline |
specifies the vertical alignment of the mstack with respect to its environment.
The legal values and their meanings are the same as that for mtable 's
align attribute.
|
||
stackalign | "left" | "center" | "right" | "decimalpoint" | decimalpoint |
specifies which column is used to horizontally align the rows.
For "left", rows are aligned flush on the left;
similarly for "right", rows are flush on the right;
for "center", the middle column (or to the right of the middle, for an even number of columns)
is used for alignment.
Rows with non-zero position , or affected by a shift ,
are treated as if the
requisite number of empty columns were added on the appropriate side;
see Section 3.6.3 Group Rows with Similiar Positions <msgroup> and Section 3.6.4 Rows in Elementary Math <msrow> .
For "decimalpoint", the column used is the left-most column in each
row that contains the decimalpoint character specified
using the decimalpoint attribute of mstyle (default ".").
If there is no decimalpoint character in the row, an implied decimal is assumed on the right of the first number in the row;
See "decimalpoint" for a discussion
of "decimalpoint".
|
||
charalign | "left" | "center" | "right" | right |
specifies the horizontal alignment of digits within a column.
If the content is larger than the column width, then it overflows the opposite side from the alignment.
For example, for "right", the content will overflow on the left side; for center,
it overflows on both sides.
This excess does not participate in the column width calculation, nor does it participate
in the overall width of the mstack .
In these cases, authors should take care to avoid collisions between column overflows.
|
||
charspacing | length | "loose" | "medium" | "tight" | medium |
specifies the amount of space to put between each column. Larger spacing might be useful if carries are not placed above or are particularly wide. The keywords "loose", "medium", and "tight" automatically adjust spacing to when carries or other entries in a column are wide. The three values allow authors to some flexibility in choosing what the layout looks like without having to figure out what values works well. In all cases, the spacing between columns is a fixed amount and does not vary between different columns. |
<mlongdiv>
Long division notation varies quite a bit around the world,
although the heart of the notation is often similar.
mlongdiv
is similar to mstack
and used to layout long division.
The first two children of mlongdiv
are the divisor and the result of the division, in that order.
The remaining children are treated as if they were children of mstack
.
The placement of these and the lines and separators used to display long division are controlled
by the longdivstyle
attribute.
The result or divisor may be an elementary math element or may be none
.
In particular, if msgroup
is used,
the elements in that group may or may not form their own mstack or be part of the dividend's mstack
,
depending upon the value of the longdivstyle
attribute.
For example, in the US style for division,
the result is treated as part of the dividend's mstack
, but divisor is not.
MathML does not specify when the result and divisor form their own mstack
,
nor does it specify what should happen if msline
or other elementary math elements
are used for the result or divisor and they do not participate in the dividend's mstack
layout.
In the remainder of this section on elementary math, anything that is said about mstack
applies
to mlongdiv
unless stated otherwise.
mlongdiv
elements accept all of the attributes that mstack
elements
accept (including those specified in Section 3.1.10 Mathematics style attributes common to presentation elements), along with the attribute listed below.
The values allowed for longdivstyle
are open-ended.
Conforming renderers may ignore any value they do not handle,
although renderers are encouraged to render as many of the values listed below as possible.
Any rules drawn as part of division layout should be drawn using the color specified by
mathcolor
.
Name | values | default |
---|---|---|
longdivstyle | "lefttop" | "stackedrightright" | "mediumstackedrightright" | "shortstackedrightright" | "righttop" | "left/\right" | "left)(right" | ":right=right" | "stackedleftleft" | "stackedleftlinetop" | lefttop |
Controls the style of the long division layout. The names are meant as a rough mnemonic that describes the position of the divisor and result in relation to the dividend. |
See Section 3.6.8.3 Long Division for examples of how these notations are drawn. The values listed above are used for long division notations in different countries around the world:
<msgroup>
msgroup
is used to group rows inside of the mstack
and mlongdiv
elements
that have a similar position relative to the alignment of stack.
If not explicitly given, the children representing the stack in mstack
and mlongdiv
are treated as if they are implicitly surrounded by an msgroup
element.
msgroup
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
position | integer | 0 |
specifies the horizontal position of the rows within this group relative
the position determined by the containing msgroup (according to its
position and shift attributes).
The resulting position value is relative to the column specified by stackalign of the containing mstack or mlongdiv .
Positive values move each row towards the tens digit,
like multiplying by a power of 10,
effectively padding with empty columns on the right;
negative values move towards the ones digit,
effectively padding on the left.
The decimal point is counted as a column and should be taken into account for negative values.
|
||
shift | integer | 0 |
specifies an incremental shift of position for successive children (rows or groups) within this group. The value is interpreted as with position, but specifies the position of each child (except the first) with respect to the previous child in the group. |
<msrow>
An msrow
represents a row in an mstack
.
In most cases it is implied by the context, but is useful
explicitly for putting multiple elements in a single row,
such as when placing an operator "+" or "-" alongside a number
within an addition or subtraction.
If an mn
element is a child of msrow
(whether implicit or not), then the number is split into its digits
and the digits are placed into successive columns.
Any other element, with the exception of mstyle
is treated effectively
as a single digit occupying the next column.
An mstyle
is treated as if its children were
directly the children of the msrow
, but with their style affected
by the attributes of the mstyle
.
The empty element none
may be used to create an empty column.
Note that a row is considered primarily as if it were a number,
which are always displayed left-to-right,
and so the directionality used to display the columns is always left-to-right;
textual bidirectionality within token elements (other than mn
) still applies,
as does the overall directionality within any children of the msrow
(which end up treated as single digits);
see Section 3.1.5 Directionality.
msrow
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
position | integer | 0 |
specifies the horizontal position of the rows within this group relative
the position determined by the containing msgroup (according to its
position and shift attributes).
The resulting position value is relative to the column specified by stackalign of the containing mstack or mlongdiv .
Positive values move each row towards the tens digit,
like multiplying by a power of 10,
effectively padding with empty columns on the right;
negative values move towards the ones digit,
effectively padding on the left.
The decimal point is counted as a column and should be taken into account for negative values.
|
<mscarries>
The mscarries
element is used for various annotations such as carries, borrows, and crossouts that occur in elementary math.
The children are associated with elements in the following row of the mstack
.
It is an error for mscarries
to be the last element of an mstack
or mlongdiv
element. Each child of the mscarries
applies to the same column in the following row.
As these annotations are used to adorn what are treated as
numbers, the attachment of carries to columns proceeds from left-to-right;
The overall directionality does not apply to the ordering of the carries,
although it may apply to the contents of each carry;
see Section 3.1.5 Directionality.
Each child of mscarries
other than mscarry
or none
is
treated as if implicitly surrounded by mscarry
;
the element none
is used when no carry for a particular column is needed.
The mscarries
element sets displaystyle
to "false", and increments scriptlevel
by 1, so the children are
typically displayed in a smaller font. (See Section 3.1.6 Displaystyle and Scriptlevel.)
It also changes the default value of scriptsizemultiplier
.
The effect is that the inherited value of
scriptsizemultiplier
should still override the default value,
but the default value, inside mscarries
, should be "0.6".
scriptsizemultiplier
can be set on the mscarries
element,
and the value should override the inherited value as usual.
If two rows of carries are adjacent to each other,
the first row of carries annotates the second (following) row as if the second row had
location
="n".
This means that the second row, even if it does not draw,
visually uses some (undefined by this specification) amount of space when displayed.
mscarries
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
position | integer | 0 |
specifies the horizontal position of the rows within this group relative
the position determined by the containing msgroup (according to its
position and shift attributes).
The resulting position value is relative to the column specified by stackalign of the containing mstack or mlongdiv .
The interpretation of the value is the same as position for msgroup or msrow ,
but it alters the association of each carry with the column below.
For example, position =1 would cause the rightmost carry to be associated with
the second digit column from the right.
|
||
location | "w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw" | n |
specifies the location of the carry or borrow relative to the character below it in the associated column. Compass directions are used for the values; the default is to place the carry above the character. | ||
crossout | ("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")* | none |
specifies how the column content below each carry is "crossed out";
one or more values may be given and all values are drawn.
If "none" is given with other values, it is ignored.
See Section 3.6.8 Elementary Math Examples for examples of the different values.
The crossout is only applied for columns which have a corresponding
mscarry .
The crossouts should be drawn using the color specified by mathcolor .
|
||
scriptsizemultiplier | number | inherited (0.6) |
specifies the factor to change the font size by.
See Section 3.1.6 Displaystyle and Scriptlevel for a description of how this works with the scriptsize attribute.
|
<mscarry>
mscarry
is used inside of mscarries
to
represent the carry for an individual column.
A carry is treated as if its width were zero; it does not participate in
the calculation of the width of its corresponding column;
as such, it may extend beyond the column boundaries.
Although it is usually implied, the element may be used explicitly to override the
location
and/or crossout
attributes of
the containing mscarries
.
It may also be useful with none
as its content in order
to display no actual carry, but still enable a crossout
due to the enclosing mscarries
to be drawn for the given column.
The mscarry
element accepts the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
Name | values | default |
---|---|---|
location | "w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw" | inherited |
specifies the location of the carry or borrow relative to the character in the corresponding column in the row below it. Compass directions are used for the values. | ||
crossout | ("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")* | inherited |
specifies how the column content associated with the carry is "crossed out";
one or more values may be given and all values are drawn.
If "none" is given with other values, it is essentially ignored.
The crossout should be drawn using the color specified by mathcolor .
|
<msline/>
msline
draws a horizontal line inside of a mstack
element.
The position, length, and thickness of the line are specified as attributes.
If the length is specified, the line is positioned and drawn as if it were a number with the given number of digits.
msline
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
The line should be drawn using the color specified by mathcolor
.
Name | values | default |
---|---|---|
position | integer | 0 |
specifies the horizontal position of the rows within this group relative
the position determined by the containing msgroup (according to its
position and shift attributes).
The resulting position value is relative to the column specified by stackalign of the containing mstack or mlongdiv .
Positive values moves towards the tens digit (like multiplying by a power of 10);
negative values moves towards the ones digit.
The decimal point is counted as a column and should be taken into account for negative values.
Note that since the default line length spans the entire mstack ,
the position has no effect unless the length is specified as non-zero.
|
||
length | unsigned-integer | 0 |
Specifies the the number of columns that should be spanned by the line.
A value of '0' (the default) means that all columns in
the row are spanned (in which case position and stackalign have no effect).
|
||
leftoverhang | length | 0 |
Specifies an extra amount that the line should overhang on the left of the leftmost column spanned by the line. | ||
rightoverhang | length | 0 |
Specifies an extra amount that the line should overhang on the right of the rightmost column spanned by the line. | ||
mslinethickness | length | "thin" | "medium" | "thick" | medium |
Specifies how thick the line should be drawn.
The line should have height=0, and depth=mslinethickness so that the top
of the msline is on the baseline of the surrounding context (if any).
(See Section 3.3.2 Fractions <mfrac> for discussion of the thickness keywords
"medium", "thin" and "thick".)
|
Two-dimensional addition, subtraction, and multiplication typically involve numbers, carrries/borrows, lines, and the sign of the operation.
Notice that the msline
spans all of the columns and that none
is used to make the "+" appear to the left of all of the operands.
The MathML for this is:
<mstack> <mn>424</mn> <msrow> <mo>+</mo> <none/> <mn>33</mn> </msrow> <msline/> </mstack> |
Here is an example with the operator on the right. Placing the operator on the right is standard in the Netherlands and some other countries. Notice that although there are a total of four columns in the example, because the default alignment is on the implied decimal point to the right of the numbers, it is not necessary to pad any row.
<mstack> <mn>123</mn> <msrow> <mn>456</mn> <mo>+</mo> </msrow> <msline/> <mn>579</mn> </mstack> |
Because the default alignment is placed to the right of number, the numbers align properly and none of the rows need to be shifted.
The following two examples illustrate the use of mscarries
,
mscarry
and using none
to fill in a column.
The examples illustrate two different ways of displaying a borrow.
|
|
The MathML for the first example is:
<mstack> <mscarries crossout='updiagonalstrike'> <mn>2</mn> <mn>12</mn> <mscarry crossout='none'> <none/> </mscarry> </mscarries> <mn>2,327</mn> <msrow> <mo>-</mo> <mn> 1,156</mn> </msrow> <msline/> <mn>1,171</mn> </mstack> |
The MathML for the second example uses mscarry
because a crossout should only happen on a single column:
<mstack> <mscarries location='nw'> <none/> <mscarry crossout='updiagonalstrike' location='n'> <mn>2</mn> </mscarry> <mn>1</mn> <none/> </mscarries> <mn>2,327</mn> <msrow> <mo>-</mo> <mn> 1,156</mn> </msrow> <msline/> <mn>1,171</mn> </mstack> |
Here is an example of subtraction where there is a borrow with multiple digits in a single column and a cross out. The borrowed amount is underlined (the example is from a Swedish source):
There are two things to notice.
The first is that menclose
is used in the carry and that none
is used for
the empty element so that mscarry
can be used to create a crossout.
<mstack> <mscarries> <mscarry crossout='updiagonalstrike'><none/></mscarry> <menclose notation='bottom'> <mn>10</mn> </menclose> </mscarries> <mn>52</mn> <msrow> <mo>-</mo> <mn> 7</mn> </msrow> <msline/> <mn>45</mn> </mstack> |
Below is a simple multiplication example that illustrates the use of msgroup
and
the shift
attribute. The first msgroup
does nothing.
The second msgroup
could also be removed, but msrow
would be needed for its second and third children.
They would set the position
or shift
attributes, or would add none
elements.
<mstack> <msgroup> <mn>123</mn> <msrow><mo>×<!--MULTIPLICATION SIGN--></mo><mn>321</mn></msrow> </msgroup> <msline/> <msgroup shift="1"> <mn>123</mn> <mn>246</mn> <mn>369</mn> </msgroup> <msline/> </mstack> |
This example has multiple rows of carries. It also (somewhat artificially) includes commas (",") as digit separators. The encoding includes these separators in the spacing attribute value, along non-ASCII values.
<mstack> <mscarries><mn>1</mn><mn>1</mn><none/></mscarries> <mscarries><mn>1</mn><mn>1</mn><none/></mscarries> <mn>1,234</mn> <msrow><mo>×<!--MULTIPLICATION SIGN--></mo><mn>4,321</mn></msrow> <msline/> <mscarries position='2'> <mn>1</mn> <none/> <mn>1</mn> <mn>1</mn> <mn>1</mn> <none/> <mn>1</mn> </mscarries> <msgroup shift="1"> <mn>1,234</mn> <mn>24,68</mn> <mn>370,2</mn> <msrow position="1"> <mn>4,936</mn> </msrow> </msgroup> <msline/> <mn>5,332,114</mn> </mstack> |
The notation used for long division varies considerably among countries. Most notations share the common characteristics of aligning intermediate results and drawing lines for the operands to be subtracted. Minus signs are sometimes shown for the intermediate calculations, and sometimes they are not. The line that is drawn varies in length depending upon the notation. The most apparent difference among the notations is that the position of the divisor varies, as does the location of the quotient, remainder, and intermediate terms.
The layout used is controlled by the longdivstyle
attribute. Below are examples for the values listed in Section 3.6.2.2 Attributes
"lefttop" | "stackedrightright" | "mediumstackedrightright" | "shortstackedrightright" | "righttop" |
---|---|---|---|---|
|
|
|
|
|
"left/\right" | "left)(right" | ":right=right" | "stackedleftleft" | "stackedleftlinetop" |
---|---|---|---|---|
|
|
|
|
|
The MathML for the first example is shown below. It illustrates the use of nested msgroup
s and how the position
is calculated in those usages.
<mlongdiv longdivstyle="lefttop"> <mn> 3 </mn> <mn> 435.3</mn> <mn> 1306</mn> <msgroup position="2" shift="-1"> <msgroup> <mn> 12</mn> <msline length="2"/> </msgroup> <msgroup> <mn> 10</mn> <mn> 9</mn> <msline length="2"/> </msgroup> <msgroup> <mn> 16</mn> <mn> 15</mn> <msline length="2"/> <mn> 1.0</mn> <!-- aligns on '.', not the right edge ('0') --> </msgroup> <msgroup position='-1'> <!-- extra shift to move to the right of the "." --> <mn> 9</mn> <msline length="3"/> <mn> 1</mn> </msgroup> </msgroup> </mlongdiv> |
With the exception of the last example,
the encodings for the other examples are the same except that the values for
longdivstyle
differ and that a "," is used instead of a "." for the decimal point.
For the last example, the only difference from the other examples besides a different value for
longdivstyle
is that Arabic numerals have been used in place of Latin numerals,
as shown below.
<mstyle decimalpoint="٫"><!--ARABIC DECIMAL SEPARATOR--> <mlongdiv longdivstyle="stackedleftlinetop"> <mn> ٣<!--ARABIC-INDIC DIGIT THREE--> </mn> <mn> ٤<!--ARABIC-INDIC DIGIT FOUR-->٣<!--ARABIC-INDIC DIGIT THREE-->٥<!--ARABIC-INDIC DIGIT FIVE-->٫<!--ARABIC DECIMAL SEPARATOR-->٣<!--ARABIC-INDIC DIGIT THREE--></mn> <mn> ١<!--ARABIC-INDIC DIGIT ONE-->٣<!--ARABIC-INDIC DIGIT THREE-->٠<!--ARABIC-INDIC DIGIT ZERO-->٦<!--ARABIC-INDIC DIGIT SIX--></mn> <msgroup position="2" shift="-1"> <msgroup> <mn> ١<!--ARABIC-INDIC DIGIT ONE-->٢<!--ARABIC-INDIC DIGIT TWO--></mn> <msline length="2"/> </msgroup> <msgroup> <mn> ١<!--ARABIC-INDIC DIGIT ONE-->٠<!--ARABIC-INDIC DIGIT ZERO--></mn> <mn> ٩<!--ARABIC-INDIC DIGIT NINE--></mn> <msline length="2"/> </msgroup> <msgroup> <mn> ١<!--ARABIC-INDIC DIGIT ONE-->٦<!--ARABIC-INDIC DIGIT SIX--></mn> <mn> ١<!--ARABIC-INDIC DIGIT ONE-->٥<!--ARABIC-INDIC DIGIT FIVE--></mn> <msline length="2"/> <mn> ١<!--ARABIC-INDIC DIGIT ONE-->٫<!--ARABIC DECIMAL SEPARATOR-->٠<!--ARABIC-INDIC DIGIT ZERO--></mn> </msgroup> <msgroup position='-1'> <mn> ٩<!--ARABIC-INDIC DIGIT NINE--></mn> <msline length="3"/> <mn> ١<!--ARABIC-INDIC DIGIT ONE--></mn> </msgroup> </msgroup> </mlongdiv> </mstyle> |
Decimal numbers that have digits that repeat infinitely such as 1/3 (.3333...) are represented using several notations. One common notation is to put a horizontal line over the digits that repeat (in Portugal an underline is used). Another notation involves putting dots over the digits that repeat. These notations are shown below:
The MathML for these involves using mstack
, msrow
, and msline
in a straightforward manner. The MathML for the preceding examples above is given below.
<mstack stackalign="right"> <msline length="1"/> <mn> 0.3333 </mn> </mstack> |
<mstack stackalign="right"> <msline length="6"/> <mn> 0.142857 </mn> </mstack> |
<mstack stackalign="right"> <mn> 0.142857 </mn> <msline length="6"/> </mstack> |
<mstack stackalign="right"> <msrow> <mo>.</mo> <none/><none/><none/><none/> <mo>.</mo> </msrow> <mn> 0.142857 </mn> </mstack> |
<maction>
To provide a mechanism for binding actions to expressions, MathML
provides the maction
element. This element accepts any
number of sub-expressions as arguments and the type of action that should happen
is controlled by the actiontype
attribute.
Only three actions are predefined by MathML, but the list of possible actions is open.
Additional predefined actions may be added in future versions of MathML.
Linking to other elements, either locally within the math
element or to some URL,
is not handled by maction
.
Instead, it is handled by adding a link directly on a MathML element as specified in
Section 6.4.4 Linking.
maction
elements accept the attributes listed
below in addition to those specified in Section 3.1.10 Mathematics style attributes common to presentation elements.
By default, MathML applications that do not recognize the specified
actiontype
should render the selected sub-expression as
defined below. If no selected sub-expression exists, it is a MathML
error; the appropriate rendering in that case is as described in
Section 2.3.2 Handling of Errors.
Name | values | default |
---|---|---|
actiontype | string | required |
Specifies what should happen for this element. The values allowed are open-ended. Conforming renderers may ignore any value they do not handle, although renderers are encouraged to render the values listed below. | ||
selection | positive-integer | 1 |
Specifies which child should be used for viewing. Its value should be between 1 and the number of
children of the element. The specified child is referred to as the "selected sub-expression" of the
maction element. If the value specified is out of range, it is an error. When the
selection attribute is not specified (including for
action types for which it makes no sense), its default value is 1, so
the selected sub-expression will be the first sub-expression.
|
If a MathML application responds to a user command to copy a MathML sub-expression to
the environment's "clipboard" (see Section 6.3 Transferring MathML), any maction
elements present in what is copied should
be given selection
values that correspond to their selection
state in the MathML rendering at the time of the copy command.
When a MathML application receives a mouse event that may be processed by two or more nested maction elements, the innermost maction element of each action type should respond to the event.
The meanings of the various actiontype
values is given below.
Note that not all renderers support all of the actiontype
values, and that the allowed values are open-ended.
selection
value,
wrapping back to 1 when it reaches the last child.
Typical uses would be for exercises in education, ellipses in long
computer algebra output, or to illustrate alternate notations. Note
that the expressions may be of significantly different size, so that
size negotiation with the browser may be desirable. If size
negotiation is not available, scrolling, elision, panning, or some
other method may be necessary to allow full viewing.
mtext
element in most circumstances.
For non-mtext
messages, renderers might provide a
natural language translation of the markup, but this is not
required.
mtext
element in most circumstances.
For non-mtext
messages,
renderers may provide a natural language translation of the markup if
full MathML rendering is not practical, but this is not
required.
maction
is replaced by what is entered, pasted, etc.
MathML does not restrict what is allowed as input, nor does it require an editor to allow arbitrary input.
Some renderers/editors may restrict the input to simple (linear) text.
The actiontype
values are open-ended. If another value is given and it requires additional attributes, the attributes must be in a different
namespace
in XML;
in HTML the attributes must begin with "data-".
An XML example is shown below:
my:color
attributes
might change the color of the characters in the presentation, while the
my:background
attribute might change the color of the background
behind the characters.
MathML uses the semantics
element to allow specifying semantic annotations to
presentation MathML elements; these can be content MathML or other notations. As such,
semantics
should be considered part of both presentation MathML and content
MathML. All MathML processors should process the semantics
element, even if they
only process one of those subsets.
In semantic annotations a presentation MathML expression is typically the first child
of the semantics
element. However, it can also be given inside of an
annotation-xml
element inside the semantics
element. If it is part of an
annotation-xml
element, then
encoding
="application/mathml-presentation+xml" or
encoding
="MathML-Presentation" may be used and presentation
MathML processors should use this value for the presentation.
See Section 5.1 Annotation Framework for more details about the
semantics
and annotation-xml
elements.
The intent of Content Markup is to provide an explicit encoding of the underlying mathematical meaning of an expression, rather than any particular rendering for the expression. Mathematics is distinguished both by its use of rigorous formal logic to define and analyze mathematical concepts, and by the use of a (relatively) formal notational system to represent and communicate those concepts. However, mathematics and its presentation should not be viewed as one and the same thing. Mathematical notation, though more rigorous than natural language, is nonetheless at times ambiguous, context-dependent, and varies from community to community. In some cases, heuristics may adequately infer mathematical semantics from mathematical notation. But in many others cases, it is preferable to work directly with the underlying, formal, mathematical objects. Content Markup provides a rigorous, extensible semantic framework and a markup language for this purpose.
The difficulties in inferring semantics from a presentation stem
from the fact that there are many to one mappings from presentation to
semantics and vice versa. For example the mathematical construct
"H multiplied by e" is often
encoded using an explicit operator as in
H × e. In different
presentational contexts, the multiplication operator might be
invisible "H e", or rendered
as the spoken word "times". Generally, many different
presentations are possible depending on the context and style
preferences of the author or reader. Thus, given
"H e" out of context it may be
impossible to decide if this is the name of a chemical or a
mathematical product of two variables H and
e. Mathematical presentation also varies across cultures
and geographical regions. For example, many notations for long
division are in use in different parts of the world today. Notations
may lose currency, for example the use of musical sharp and flat
symbols to denote maxima and minima [Chaundy1954]. A
notation in use in 1644 for the multiplication mentioned above was
He [Cajori1928].
By encoding the underlying mathematical structure explicitly, without regard to how it is presented aurally or visually, it is possible to interchange information more precisely between systems that semantically process mathematical objects. In the trivial example above, such a system could substitute values for the variables H and e and evaluate the result. Important application areas include computer algebra systems, automatic reasoning system, industrial and scientific applications, multi-lingual translation systems, mathematical search, and interactive textbooks.
The organization of this chapter is as follows. In Section 4.2 Content MathML Elements Encoding Expression Structure, a core collection of elements comprising Strict Content Markup are described. Strict Content Markup is sufficient to encode general expression trees in a semantically rigorous way. It is in one-to-one correspondence with OpenMath element set. OpenMath is a standard for representing formal mathematical objects and semantics through the use of extensible Content Dictionaries. Strict Content Markup defines a mechanism for associating precise mathematical semantics with expression trees by referencing OpenMath Content Dictionaries. The next two sections introduce markup that is more convenient than Strict markup for some purposes, somewhat less formal and verbose. In Section 4.3 Content MathML for Specific Structures, markup is introduced for representing a small number of mathematical idioms, such as limits on integrals, sums and product. These constructs may all be rewritten as Strict Content Markup expressions, and rules for doing so are given. In Section 4.4 Content MathML for Specific Operators and Constants, elements are introduced for many common function, operators and constants. This section contains many examples, including equivalent Strict Content expressions. In Section 4.5 Deprecated Content Elements, elements from MathML 1 and 2 whose use is now discouraged are listed. Finally, Section 4.6 The Strict Content MathML Transformation summarizes the algorithm for translating arbitrary Content Markup into Strict Content Markup. It collects together in sequence all the rewrite rules introduced throughout the rest of the chapter.
Content MathML represents mathematical objects as expression trees. The notion of constructing a general expression tree is e.g. that of applying an operator to sub-objects. For example, the sum "x+y" can be thought of as an application of the addition operator to two arguments x and y. And the expression "cos(π)" as the application of the cosine function to the number π.
As a general rule, the terminal nodes in the tree represent basic mathematical objects such as numbers, variables, arithmetic operations and so on. The internal nodes in the tree represent function application or other mathematical constructions that build up a compound objects. Function application provides the most important example; an internal node might represent the application of a function to several arguments, which are themselves represented by the nodes underneath the internal node.
The semantics of general mathematical expressions is not a matter of consensus. It would be an enormous job to systematically codify most of mathematics – a task that can never be complete. Instead, MathML makes explicit a relatively small number of commonplace mathematical constructs, chosen carefully to be sufficient in a large number of applications. In addition, it provides a mechanism for referring to mathematical concepts outside of the base collection, allowing them to be represented, as well.
The base set of content elements is chosen to be adequate for simple coding of most of the formulas used from kindergarten to the end of high school in the United States, and probably beyond through the first two years of college, that is up to A-Level or Baccalaureate level in Europe.
While the primary role of the MathML content element set is to directly encode the mathematical structure of expressions independent of the notation used to present the objects, rendering issues cannot be ignored. There are different approaches for rendering Content MathML formulae, ranging from native implementations of the MathML elements to declarative notation definitions, to XSLT style sheets. Because rendering requirements for Content MathML vary widely, MathML 3 does not provide a normative specification for rendering. Instead, typical renderings are suggested by way of examples.
In MathML 3, a subset, or profile, of Content MathML is defined: Strict Content MathML. This uses a minimal, but sufficient, set of elements to represent the meaning of a mathematical expression in a uniform structure, while the full Content MathML grammar is backward compatible with MathML 2.0, and generally tries to strike a more pragmatic balance between verbosity and formality.
Content MathML provides a large number of predefined functions
encoded as empty elements (e.g. sin
, log
, etc.)
and a variety of constructs for forming compound objects
(e.g. set
, interval
, etc.). By contrast, Strict
Content MathML uses a single element (csymbol
) with an
attribute pointing to an external definition in extensible content
dictionaries to represent all functions, and uses only
apply
and bind
for building up compound
objects. The token elements such as ci
and cn
are
also considered part of Strict Content MathML, but with a more
restricted set of attributes and with content restricted to
text.
Strict Content MathML is designed to be compatible with OpenMath (in fact it is an XML encoding of OpenMath Objects in the sense of [OpenMath2004]). OpenMath is a standard for representing formal mathematical objects and semantics through the use of extensible Content Dictionaries. The table below gives an element-by-element correspondence between the OpenMath XML encoding of OpenMath objects and Strict Content MathML.
Strict Content MathML | OpenMath |
---|---|
cn |
OMI , OMF |
csymbol |
OMS |
ci |
OMV |
cs |
OMSTR |
apply |
OMA |
bind |
OMBIND |
bvar |
OMBVAR |
share |
OMR |
semantics |
OMATTR |
annotation ,
annotation-xml |
OMATP , OMFOREIGN |
cerror |
OME |
cbytes |
OMB |
In MathML 3, formal semantics Content MathML expressions are given by specifying equivalent Strict Content MathML expressions. Since Strict Content MathML expressions all have carefully-defined semantics given in terms of OpenMath Content Dictionaries, all Content MathML expressions inherit well-defined semantics in this way. To make the correspondence exact, an algorithm is given in terms of transformation rules that are applied to rewrite non-Strict MathML constructs into a strict equivalents. The individual rules are introduced in context throughout the chapter. In Section 4.6 The Strict Content MathML Transformation, the algorithm as a whole is described.
As most transformation rules relate to classes of MathML elements that have similar argument structure, they are introduced in Section 4.3.4 Operator Classes where these classes are defined. Some special case rules for specific elements are given in Section Section 4.4 Content MathML for Specific Operators and Constants. Transformations in Section 4.2 Content MathML Elements Encoding Expression Structure concern non-Strict usages of the core Content MathML elements, those in Section 4.3 Content MathML for Specific Structures concern the rewriting of some additional structures not directly supported in Strict Content MathML.
The full algorithm described inSection 4.6 The Strict Content MathML Transformation is complete in the sense that it gives every Content MathML expression a specific meaning in terms of a Strict Content MathML expression. This means it has to give specific strict interpretations to some expressions whose meaning was insufficiently specified in MathML2. The intention of this algorithm is to be faithful to mathematical intuitions. However edge cases may remain where the normative interpretation of the algorithm may break earlier intuitions.
A conformant MathML processor need not implement this transformation. The existence of these transformation rules does not imply that a system must treat equivalent expressions identically. In particular systems may give different presentation renderings for expressions that the transformation rules imply are mathematically equivalent.
Due to the nature of mathematics, any method for formalizing
the meaning of the mathematical expressions must be
extensible. The key to extensibility is the ability to define
new functions and other symbols to expand the terrain of
mathematical discourse. To do this, two things are required: a
mechanism for representing symbols not already defined by
Content MathML, and a means of associating a specific
mathematical meaning with them in an unambiguous way. In MathML
3, the csymbol
element provides the means to represent
new symbols, while Content Dictionaries are the way
in which mathematical semantics are described. The association
is accomplished via attributes of the csymbol
element
that point at a definition in a CD. The syntax and usage of
these attributes are described in detail in Section 4.2.3 Content Symbols <csymbol>
.
Content Dictionaries are structured documents for the definition of mathematical concepts; see the OpenMath standard, [OpenMath2004]. To maximize modularity and reuse, a Content Dictionary typically contains a relatively small collection of definitions for closely related concepts. The OpenMath Society maintains a large set of public Content Dictionaries including the MathML CD group that including contains definitions for all pre-defined symbols in MathML. There is a process for contributing privately developed CDs to the OpenMath Society repository to facilitate discovery and reuse. MathML 3 does not require CDs be publicly available, though in most situations the goals of semantic markup will be best served by referencing public CDs available to all user agents.
In the text below, descriptions of semantics for predefined MathML symbols refer to the Content Dictionaries developed by the OpenMath Society in conjunction with the W3C Math Working Group. It is important to note, however, that this information is informative, and not normative. In general, the precise mathematical semantics of predefined symbols are not not fully specified by the MathML 3 Recommendation, and the only normative statements about symbol semantics are those present in the text of this chapter. The semantic definitions provided by the OpenMath Content CDs are intended to be sufficient for most applications, and are generally compatible with the semantics specified for analogous constructs in the MathML 2.0 Recommendation. However, in contexts where highly precise semantics are required (e.g. communication between computer algebra systems, within formal systems such as theorem provers, etc.) it is the responsibility of the relevant community of practice to verify, extend or replace definitions provided by OpenMath CDs as appropriate.
The basic building blocks of Content MathML expressions are numbers, identifiers and symbols. These building blocks are combined using function applications and binding operators. It is important to have a basic understanding of these key mathematical concepts, and how they are reflected in the design of Content MathML. For the convenience of the reader, these concepts are reviewed here.
In the expression "x+y",
x is a mathematical variable, meaning an identifier that
represents a quantity with no fixed value. It may have other
properties, such as being an integer, but its value is not a fixed
property. By contrast, the plus sign is an identifier that represents a
fixed and externally defined object, namely the addition function.
Such an identifier is termed a symbol, to distinguish it
from a variable. Common elementary functions and operators all have
fixed, external definitions, and are hence symbols. Content MathML
uses the ci
element to represent variables, and the
csymbol
to represent symbols.
The most fundamental way in which symbols and variables are
combined is function application. Content MathML makes a crucial
semantic distinction between a function itself (a symbol such as the
sine function, or a variable such as f) and the result of
applying the function to arguments. The apply
element groups the function
with its arguments syntactically, and represents the expression
resulting from applying that function to its arguments.
Mathematically, variables are divided into bound and
free variables. Bound variables are variables that are
assigned a special role by a binding operator within a certain scope.
For example, the index variable within a summation is a bound
variable. They can be characterized as variables with the property
that they can be renamed consistently throughout the binding scope
without changing the underlying meaning of the expression. Variables
that are not bound are termed free variables.
Because the logical distinction between bound and free variables is
important for well-defined semantics, Content MathML differentiates
between the application of a function to a free variable,
e.g. f(x) and the operation of binding a
variable within a scope. The bind
element is used the delineate
the binding scope, and group the binding operator with its bound
variables, which are indicated by the bvar
element.
In Strict Content markup, the bind
element is the only way
of performing variable binding. In non-Strict usage, however, markup
is provided that more closely resembles well-known idiomatic
notations, such as the "limit" notations for sums and
integrals. These constructs often implicitly bind variables, such as
the variable of integration, or the index variable in a sum. MathML
terms the elements used to represent the auxiliary data such as limits
required by these constructions qualifier elements.
Expressions involving qualifiers follow one of a small number of idiomatic patterns, each of which applies to class of similar binding operators. For example, sums and products are in the same class because they use index variables following the same pattern. The Content MathML operator classes are described in detail in Section 4.3.4 Operator Classes.
Each Content MathML element is described in a section below that
begins with a table summarizing the key information about the element.
For elements that have different Strict and non-Strict
usage, these syntax tables are divided to clearly separate the two cases. The element's content
model is given in the Content row, linked to the MathML
Schema in Appendix A Parsing MathML. The Attributes, and
Attribute Values rows similarly link to the schema. Where
applicable, the Class row specifies the operator class, which
indicate how many arguments the operator represented by this element
takes, and also in many cases determines the mapping to Strict Content
MathML, as described in Section 4.3.4 Operator Classes. Finally,
the Qualifiers row clarifies whether the operator takes
qualifiers and if so, which. Note Class
and Qualifiers specify how many siblings may follow
the operator element in an apply
, or the children of the
element for container elements; see Section 4.2.5 Function Application <apply>
and Section 4.3.3 Qualifiers for
details).
In this section we will present the elements for encoding the structure of content MathML expressions. These elements are the only ones used for the Strict Content MathML encoding. Concretely, we have
basic expressions, i.e. Numbers, string literals, encoded bytes, Symbols, and Identifiers.
derived expressions, i.e. function applications and binding expressions, and
Full Content MathML allows further elements presented in Section 4.3 Content MathML for Specific Structures and Section 4.4 Content MathML for Specific Operators and Constants, and allows a richer content model presented in this section. Differences in Strict and non-Strict usage of are highlighted in the sections discussing each of the Strict element below.
<cn>
Schema Fragment (Strict) | Schema Fragment (Full) | |||
---|---|---|---|---|
Class | Cn | Cn | ||
Attributes | CommonAtt, type | CommonAtt, DefEncAtt, type?, base? | ||
type Attribute Values
|
"integer" | "real" | "double" | "hexdouble" | "integer" | "real" | "double" | "hexdouble" | "e-notation" | "rational" | "complex-cartesian" | "complex-polar" | "constant" | text | default is real | |
base Attribute Values
|
integer | default is 10 | ||
Content | text | (text | mglyph | sep | PresentationExpression)* |
The cn
element is the Content MathML element used to
represent numbers. Strict Content MathML supports integers, real numbers,
and double precision floating point numbers. In these types of numbers,
the content of cn
is text. Additionally, cn
supports rational numbers and complex numbers in which the different
parts are separated by use of the sep
element. Constructs
using sep
may be rewritten in Strict Content MathML as
constructs using apply
as described below.
The type
attribute specifies which kind of number is
represented in the cn
element. The default value is
"real". Each type implies that the content be of
a certain form, as detailed below.
<cn>
,<sep/>
-Represented Numbers
The default rendering of the text content of cn
is the same as that of the Presentation element mn
, with suggested variants in the
case of attributes or sep
being used, as listed below.
<cn>
In Strict Content MathML, the type
attribute is mandatory, and may only take the values
"integer", "real", "hexdouble" or
"double":
cn
as an IEEE
double are specified by Section
3.1.2.5 of XML Schema Part 2: Datatypes Second Edition
[XMLSchemaDatatypes]. For example, -1E4, 1267.43233E12, 12.78e-2,
12 , -0, 0 and INF are all valid doubles in this format.
This type is used to directly represent the 64 bits of an IEEE 754 double-precision floating point number as a 16 digit hexadecimal number. Thus the number represents mantissa, exponent, and sign from lowest to highest bits using a least significant byte ordering. This consists of a string of 16 digits 0-9, A-F. The following example represents a NaN value. Note that certain IEEE doubles, such as the NaN in the example, cannot be represented in the lexical format for the "double" type.
<cn type="hexdouble">7F800000</cn> |
Sample Presentation
<mn>0x7F800000</mn> |
<cn>
The base
attribute is used to specify how the content is
to be parsed. The attribute value is a base 10 positive integer
giving the value of base in which the text content of the cn
is to be interpreted. The base
attribute should only be
used on elements with type "integer" or
"real". Its use on cn
elements of other type
is deprecated. The default value for base
is
"10".
Additional values for the type
attribute element for supporting
e-notations for real numbers, rational numbers, complex numbers and selected important
constants. As with the "integer", "real",
"double" and "hexdouble" types, each of these types
implies that the content be of a certain form. If the type
attribute is
omitted, it defaults to "real".
Integers can be represented with respect to a base different from
10: If base
is present, it specifies (in base 10) the base for the digit encoding.
Thus base
='16' specifies a hexadecimal
encoding. When base
> 10, Latin letters (A-Z, a-z) are used in
alphabetical order as digits. The case of letters used as digits is not
significant. The following example encodes the base 10 number 32736.
<cn base="16">7FE0</cn> |
Sample Presentation
<msub><mn>7FE0</mn><mn>16</mn></msub> |
When base
> 36, some integers cannot be represented using
numbers and letters alone. For example, while
<cn base="1000">10F</cn> |
arguably represents the number written in base 10 as 1,000,015, the number
written in base 10 as 1,000,037 cannot be represented using letters and
numbers alone when base
is 1000. Consequently, support
for additional characters (if any) that may be used for digits when base
> 36 is application specific.
base
attribute is present, then the digits are
interpreted as being digits computed relative to that base (in the same way as
described for type "integer").
A real number may be presented in scientific notation using this type. Such
numbers have two parts (a significand and an exponent)
separated by a <sep/>
element. The
first part is a real number, while the
second part is an integer exponent indicating a power of the base.
For example, <cn type="e-notation">12.3<sep/>5</cn>
represents 12.3 times 105. The default presentation of this example is
12.3e5. Note that this type is primarily useful for backwards compatibility with
MathML 2, and in most cases, it is preferable to use the "double"
type, if the number to be represented is in the range of IEEE doubles:
A rational number is given as two integers to be used as the numerator and
denominator of a quotient. The numerator and denominator are
separated by <sep/>
.
<cn type="rational">22<sep/>7</cn> |
Sample Presentation
<mrow><mn>22</mn><mo>/</mo><mn>7</mn></mrow> |
A complex cartesian number is given as two numbers specifying the real and
imaginary parts. The real and imaginary parts are separated
by the <sep/>
element, and each part has
the format of a real number as described above.
<cn type="complex-cartesian"> 12.3 <sep/> 5 </cn> |
Sample Presentation
<mrow> <mn>12.3</mn><mo>+</mo><mn>5</mn><mo>⁢<!--INVISIBLE TIMES--></mo><mi>i</mi> </mrow> |
A complex polar number is given as two numbers specifying
the magnitude and angle. The magnitude and angle are separated
by the <sep/>
element, and each part has
the format of a real number as described above.
<cn type="complex-polar"> 2 <sep/> 3.1415 </cn> |
Sample Presentation
<mrow> <mn>2</mn> <mo>⁢<!--INVISIBLE TIMES--></mo> <msup> <mi>e</mi> <mrow><mi>i</mi><mo>⁢<!--INVISIBLE TIMES--></mo><mn>3.1415</mn></mrow> </msup> </mrow> |
<mrow> <mi>Polar</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mn>2</mn><mn>3.1415</mn></mfenced> </mrow> |
If the value type
is "constant",
then the content should be a Unicode representation of a
well-known constant. Some important constants and their
common Unicode representations are listed below.
This cn
type is primarily for backward
compatibility with MathML 1.0. MathML 2.0 introduced many
empty elements, such as <pi/>
to
represent constants, and using these representations or
a Strict csymbol representation is preferred.
In addition to the additional values of the type attribute, the
content of cn
element can contain (in addition to the
sep
element allowed in Strict Content MathML) mglyph
elements to refer to characters not currently available in Unicode, or
a general presentation construct (see Section 3.1.9 Summary of Presentation Elements),
which is used for rendering (see Section 4.1.2 The Structure and Scope of Content MathML Expressions).
Mapping to Strict Content MathML
If a base
attribute is present, it specifies the base used for the digit
encoding of both integers. The use of base
with
"rational" numbers is deprecated.
If there are sep
children of the cn
,
then intervening text may be rewritten as cn
elements. If the cn
element containing sep
also has a base
attribute, this is copied to each
of the cn
arguments of the resulting symbol, as
shown below.
<cn type="" base=" "> <sep/> </cn> |
is rewritten to
<apply><csymbol cd=""> </csymbol> <cn type="integer" base=" "> </cn> <cn type="integer" base=" "> </cn> </apply> |
The symbol used in the result depends on the type
attribute according to the following table:
type attribute | OpenMath Symbol |
---|---|
e-notation | bigfloat |
rational | rational |
complex-cartesian | complex_cartesian |
complex-polar | complex_polar |
Note: In the case of bigfloat the symbol
takes three arguments, <cn type="integer">10</cn>
should be inserted as the second argument, denoting the base of the exponent used.
If the type
attribute has a different value,
or if there is more than one <sep/>
element,
then the intervening expressions are converted as above,
but a system-dependent choice of symbol for the head of the application must be used.
If a base attribute has been used then the resulting expression is not Strict Content MathML, and each of the arguments needs to be recursively processed.
A cn
element with a base attribute other than 10 is rewritten as follows. (A base attribute with value 10 is simply removed) .
<cn type="" base=" "> </cn> |
<apply><csymbol cd="nums1"></csymbol> <cn type="integer"> </cn> <cs> </cs> </apply> |
If the original element specified type "integer" or if there is no type attribute, but the content of the element just consists of the characters [a-zA-Z0-9] and white space then the symbol used as the head in the resulting application should be based_integer as shown. Otherwise it should be should be based_float.
In Strict Content MathML, constants should be represented using
csymbol
elements. A number of important constants are defined in the
nums1 content dictionary. An expression of the form
<cn type="constant"> </cn> |
has the Strict Content MathML equivalent
<csymbol cd="nums1"> </csymbol> |
where c2
corresponds to c
as specified in the following table.
Content | Description | OpenMath Symbol |
---|---|---|
U+03C0 (π )
|
The usual π of trigonometry: approximately 3.141592653... | pi |
U+2147 (ⅇ or ⅇ )
|
The base for natural logarithms: approximately 2.718281828... | e |
U+2148 (ⅈ or ⅈ )
|
Square root of -1 | i |
U+03B3 (γ )
|
Euler's constant: approximately 0.5772156649... | gamma |
U+221E (∞ or &infty; )
|
Infinity. Proper interpretation varies with context | infinity |
If the cn
contains Presentation MathML markup, then it may
be rewritten to Strict MathML using variants of the rules above where
the arguments of the constructor are ci
elements annotated
with the supplied Presentation MathML.
A cn
expression with non-text content of the form
<cn type=""> <sep/> </cn> |
is transformed to Strict Content MathML by rewriting it to
<apply><csymbol cd=""> </csymbol> <semantics> <ci> </ci> <annotation-xml encoding="MathML-Presentation"> </annotation-xml> </semantics> <semantics> <ci> </ci> <annotation-xml encoding="MathML-Presentation"> </annotation-xml> </semantics> </apply> |
Where the identifier names, p and q, (which have to be a text string) should be
determined from the presentation MathML content, in a system defined way, perhaps as
in the above example by taking the character data of the element ignoring any element
markup. Systems doing such rewriting should ensure that constructs using the same
Presentation MathML content are rewritten to semantics
elements using the
same ci
, and that conversely constructs that use different MathML should be
rewritten to different identifier names (even if the Presentation MathML has the same character data).
A related special case arises when a cn
element
contains character data not permitted in Strict Content MathML
usage, e.g. non-digit, alphabetic characters. Conceptually, this is
analogous to a cn
element containing a presentation
markup mtext
element, and could be rewritten accordingly.
However, since the resulting annotation would contain no additional
rendering information, such instances should be rewritten directly
as ci
elements, rather than as a semantics
construct.
<ci>
Schema Fragment (Strict) | Schema Fragment (Full) | |
---|---|---|
Class | Ci | Ci |
Attributes | CommonAtt, type? | CommonAtt, DefEncAtt, type? |
type Attribute Values
|
"integer"| "rational"| "real"| "complex"| "complex-polar"| "complex-cartesian"| "constant"| "function"| "vector"| "list"| "set"| "matrix" | string |
Qualifiers | BvarQ, DomainQ, degree, momentabout, logbase | |
Content | text | text | mglyph | PresentationExpression |
Content MathML uses the ci
element (mnemonic for "content
identifier") to construct a variable. Content identifiers
represent "mathematical variables" which have
properties, but no fixed value. For example, x and y are variables
in the expression "x+y", and the variable
x would be represented as
<ci>x</ci> |
In MathML, variables are distinguished from symbols, which have fixed, external definitions, and are represented by the csymbol element.
After white space normalization the content of a ci
element is interpreted as a
name that identifies it. Two variables are considered equal, if and only if their names
are identical and in the same scope (see Section 4.2.6 Bindings and Bound Variables <bind>
and <bvar>
for a
discussion).
<ci>
The ci
element uses the type
attribute to specify the basic type of
object that it represents. In Strict Content MathML, the set of permissible values is
"integer", "rational", "real",
"complex", "complex-polar",
"complex-cartesian", "constant", "function",
vector
, list
, set
, and matrix
. These values correspond
to the symbols
integer_type,
rational_type,
real_type,
complex_polar_type,
complex_cartesian_type,
constant_type,
fn_type,
vector_type,
list_type,
set_type, and
matrix_type in the
mathmltypes Content Dictionary: In this sense the following two expressions are considered equivalent:
<ci type="integer">n</ci> |
<semantics> <ci>n</ci> <annotation-xml cd="mathmltypes" name="type" encoding="MathML-Content"> <csymbol cd="mathmltypes">integer_type</csymbol> </annotation-xml> </semantics> |
Note that "complex" should be considered
an alias for "complex-cartesian" and rewritten to the
same complex_cartesian_type
symbol. It is perhaps a more natural type name for use with
ci
as the distinction between cartesian and polar form really
only affects the interpretation of literals encoded with cn
.
<ci>
The ci
element allows any string value for the type
attribute, in particular any of the names of the MathML container elements or their type
values.
For a more advanced treatment of types, the type
attribute is
inappropriate. Advanced types require significant structure of their own (for example,
vector(complex)) and are probably best constructed as mathematical objects and
then associated with a MathML expression through use of the semantics
element. See [MathMLTypes] for more examples.
Mapping to Strict Content MathML
In Strict Content, type attributes are represented via semantic attribution. An expression of the form
<ci type=""> </ci> |
is rewritten to
<semantics> <ci></ci> <annotation-xml cd="mathmltypes" name="type" encoding="MathML-Content"> <ci> </ci> </annotation-xml> </semantics> |
The ci
element can contain
mglyph
elements to refer to characters not currently available in Unicode, or a
general presentation construct (see Section 3.1.9 Summary of Presentation Elements), which is used for
rendering (see Section 4.1.2 The Structure and Scope of Content MathML Expressions).
A ci
expression with non-text content of the form
<ci> </ci> |
is transformed to Strict Content MathML by rewriting it to
<semantics> <ci></ci> <annotation-xml encoding="MathML-Presentation"> </annotation-xml> </semantics> |
Where the identifier name, p, (which has to be a text string) should be
determined from the presentation MathML content, in a system defined way, perhaps as
in the above example by taking the character data of the element ignoring any element
markup. Systems doing such rewriting should ensure that constructs using the same
Presentation MathML content are rewritten to semantics
elements using the
same ci
, and that conversely constructs that use different MathML should be
rewritten to different identifier names (even if the Presentation MathML has the same character data).
The following example encodes an atomic symbol that displays visually as C2 and that, for purposes of content, is treated as a single symbol
<ci> <msup><mi>C</mi><mn>2</mn></msup> </ci> |
The Strict Content MathML equivalent is
<semantics> <ci>C2</ci> <annotation-xml encoding="MathML-Presentation"> <msup><mi>C</mi><mn>2</mn></msup> </annotation-xml> </semantics> |
Sample Presentation
<msup><mi>C</mi><mn>2</mn></msup> |
If the content of a ci
element consists of Presentation MathML, that
presentation is used. If no such tagging is supplied then the text
content is rendered as if it were the content of an mi
element. If an
application supports bidirectional text rendering, then the rendering follows the
Unicode bidirectional rendering.
The type
attribute can be interpreted to
provide rendering information. For example in
<ci type="vector">V</ci> |
a renderer could display a bold V for the vector.
<csymbol>
Schema Fragment (Strict) | Schema Fragment (Full) | |
---|---|---|
Class | Csymbol | Csymbol |
Attributes | CommonAtt, cd | CommonAtt, DefEncAtt, type?, cd? |
Content | SymbolName | text | mglyph | PresentationExpression |
Qualifiers | BvarQ, DomainQ, degree, momentabout, logbase |
A csymbol
is used to refer to a specific,
mathematically-defined concept with an external definition. In the
expression "x+y", the plus sign is
a symbol since it has a specific, external definition, namely the addition function.
MathML 3 calls such an identifier a
symbol. Elementary functions and common mathematical
operators are all examples of symbols. Note that the term
"symbol" is used here in an abstract sense and has no
connection with any particular presentation of the construct on screen
or paper.
<csymbol>
The csymbol
identifies the specific mathematical concept
it represents by referencing its definition via attributes.
Conceptually, a reference to an external definition is merely a URI,
i.e. a label uniquely identifying the definition. However, to be
useful for communication between user agents, external definitions
must be shared.
For this reason, several longstanding efforts have been organized to develop systematic, public repositories of mathematical definitions. Most notable of these, the OpenMath Society repository of Content Dictionaries (CDs) is extensive, open and active. In MathML 3, OpenMath CDs are the preferred source of external definitions. In particular, the definitions of pre-defined MathML 3 operators and functions are given in terms of OpenMath CDs.
MathML 3 provides two mechanisms for referencing external definitions or content
dictionaries. The first, using the cd
attribute, follows conventions
established by OpenMath specifically for referencing CDs. This is the
form required in Strict Content MathML. The second, using the
definitionURL
attribute, is backward compatible with MathML 2, and can be used
to reference CDs or any other source of definitions that can be
identified by a URI. It is described in the following section
When referencing OpenMath CDs, the preferred method is to use the cd
attribute as follows. Abstractly, OpenMath symbol definitions are identified by a triple
of values: a symbol name, a CD name, and a CD base,
which is a URI that disambiguates CDs of the same name. To associate such a triple with a
csymbol
, the content of the csymbol
specifies the symbol name, and the
name of the Content Dictionary is given using the cd
attribute. The CD base is
determined either from the document embedding the math
element which contains the
csymbol
by a mechanism given by the embedding document format, or by system
defaults, or by the cdgroup
attribute , which is optionally specified on the
enclosing math
element; see Section 2.2.1 Attributes. In the absence
of specific information http://www.openmath.org/cd
is assumed as the CD base
for all csymbol
elements annotation
, and annotation-xml
. This
is the CD base for the collection of standard CDs maintained by the OpenMath Society.
The cdgroup
specifies a URL to an OpenMath CD Group file. For a detailed
description of the format of a CD Group file, see Section 4.4.2 (CDGroups)
in [OpenMath2004]. Conceptually, a CD group file is a list of
pairs consisting of a CD name, and a corresponding CD base. When a csymbol
references a CD name using the cd
attribute, the name is looked up in the CD
Group file, and the associated CD base value is used for that csymbol
. When a CD
Group file is specified, but a referenced CD name does not appear in the group file, or
there is an error in retrieving the group file, the referencing csymbol
is not
defined. However, the handling of the resulting error is not defined, and is the
responsibility of the user agent.
While references to external definitions are URIs, it is strongly recommended that CD files be retrievable at the location obtained by interpreting the URI as a URL. In particular, other properties of the symbol being defined may be available by inspecting the Content Dictionary specified. These include not only the symbol definition, but also examples and other formal properties. Note, however, that there are multiple encodings for OpenMath Content Dictionaries, and it is up to the user agent to correctly determine the encoding when retrieving a CD.
<csymbol>
In addition to the forms described above, the csymbol
and element can contain
mglyph
elements to refer to characters not currently available in Unicode, or a
general presentation construct (see Section 3.1.9 Summary of Presentation Elements), which is used for
rendering (see Section 4.1.2 The Structure and Scope of Content MathML Expressions). In this case, when
writing to Strict Content MathML, the csymbol should be treated as a
ci
element, and rewritten using Rewrite: ci presentation mathml.
External definitions (in OpenMath CDs or elsewhere) may also be specified directly for
a csymbol
using the definitionURL
attribute. When used to reference
OpenMath symbol definitions, the abstract triple of (symbol name, CD name, CD base) is
mapped to a fully-qualified URI as follows:
URI =
cdbase+ '/' +
cd-name+ '#' +
symbol-name
For example,
(plus, arith1, http://www.openmath.org/cd)
is mapped to
http://www.openmath.org/cd/arith1#plus
The resulting URI is specified as the value of the definitionURL
attribute.
This form of reference is useful for backwards compatibility with MathML2 and to
facilitate the use of Content MathML within URI-based frameworks (such as RDF [rdf] in the Semantic Web or OMDoc [OMDoc1.2]). Another benefit is
that the symbol name in the CD does not need to correspond to the content of the
csymbol
element. However, in general, this method results in much longer MathML
instances. Also, in situations where CDs are under development, the use of a CD Group
file allows the locations of CDs to change without a change to the markup. A third
drawback to definitionURL
is that unlike the cd
attribute, it is not
limited to referencing symbol definitions in OpenMath content dictionaries. Hence, it is
not in general possible for a user agent to automatically determine the proper
interpretation for definitionURL
values without further information about the
context and community of practice in which the MathML instance occurs.
Both the cd
and definitionURL
mechanisms of external reference
may be used within a single MathML instance. However, when both a cd
and a
definitionURL
attribute are specified on a single csymbol
, the
cd
attribute takes precedence.
Mapping to Strict Content MathML
In non-Strict usage csymbol
allows the use of
a type
attribute.
In Strict Content, type attributes are represented via semantic attribution. An expression of the form
<csymbol type=" ">symbolname</csymbol> |
is rewritten to
<semantics>
<csymbol>symbolname</csymbol>
<annotation-xml cd="mathmltypes" name="type" encoding="MathML-Content">
<ci> </ci>
</annotation-xml>
</semantics> |
If the content of a csymbol
element is tagged using presentation tags,
that presentation is used. If no such tagging is supplied then the text
content is rendered as if it were the content of an mi
element. In
particular if an application supports bidirectional text rendering, then the
rendering follows the Unicode bidirectional rendering.
<cs>
Schema Fragment (Strict) | Schema Fragment (Full) | |
---|---|---|
Class | Cs | Cs |
Attributes | CommonAtt | CommonAtt, DefEncAtt |
Content | text | text |
The cs
element encodes "string literals"
which may be used in Content MathML expressions.
The content of cs is text; no
Presentation MathML constructs are allowed even when used in
non-strict markup. Specifically, cs
may not contain
mglyph
elements, and the content does not undergo white space
normalization.
Content MathML
<set> <cs>A</cs><cs>B</cs><cs> </cs> </set> |
Sample Presentation
<mrow> <mo>{</mo> <ms>A</ms> <mo>,</mo> <ms>B</ms> <mo>,</mo> <ms> <!--NO-BREAK SPACE--> <!--NO-BREAK SPACE--></ms> <mo>}</mo> </mrow> |
<apply>
Schema Fragment (Strict) | Schema Fragment (Full) | |
---|---|---|
Class | Apply | Apply |
Attributes | CommonAtt | CommonAtt, DefEncAtt |
Content | ContExp+ | ContExp+ | (ContExp, BvarQ, Qualifier?, ContExp*) |
The most fundamental way of building a compound object in mathematics is by applying a function or an operator to some arguments.
In MathML, the apply
element is used to build an expression tree that
represents the application a function or operator to its arguments. The
resulting tree corresponds to a complete mathematical expression. Roughly
speaking, this means a piece of mathematics that could be surrounded by
parentheses or "logical brackets" without changing its meaning.
For example, (x + y) might be encoded as
<apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><ci>y</ci></apply> |
The opening and closing tags of apply
specify exactly the scope of any
operator or function. The most typical way of using apply
is simple and
recursive. Symbolically, the content model can be described as:
<apply> op [ a b ...] </apply>
where the operands a, b, ... are MathML
expression trees themselves, and op is a MathML expression tree that
represents an operator or function. Note that apply
constructs can be
nested to arbitrary depth.
An apply
may in principle have any number of operands. For example,
(x + y + z) can be encoded as
<apply><csymbol cd="arith1">plus</csymbol> <ci>x</ci> <ci>y</ci> <ci>z</ci> </apply> |
Note that MathML also allows applications without operands, e.g. to represent functions like random()
, or current-date()
.
Mathematical expressions involving a mixture of operations result in nested
occurrences of apply
. For example, a x + b
would be encoded as
<apply><csymbol cd="arith1">plus</csymbol> <apply><csymbol cd="arith1">times</csymbol> <ci>a</ci> <ci>x</ci> </apply> <ci>b</ci> </apply> |
There is no need to introduce parentheses or to resort to
operator precedence in order to parse expressions correctly. The
apply
tags provide the proper grouping for the re-use
of the expressions within other constructs. Any expression
enclosed by an apply
element is well-defined, coherent
object whose interpretation does not depend on the surrounding
context. This is in sharp contrast to presentation markup,
where the same expression may have very different meanings in
different contexts. For example, an expression with a visual
rendering such as (F+G)(x)
might be a product, as in
<apply><csymbol cd="arith1">times</csymbol> <apply><csymbol cd="arith1">plus</csymbol> <ci>F</ci> <ci>G</ci> </apply> <ci>x</ci> </apply> |
or it might indicate the application of the function F + G to the argument x. This is indicated by constructing the sum
<apply><csymbol cd="arith1">plus</csymbol><ci>F</ci><ci>G</ci></apply> |
and applying it to the argument x as in
<apply> <apply><csymbol cd="arith1">plus</csymbol> <ci>F</ci> <ci>G</ci> </apply> <ci>x</ci> </apply> |
In both cases, the interpretation of the outer apply
is
explicit and unambiguous, and does not change regardless of
where the expression is used.
The preceding example also illustrates that in an
apply
construct, both the function and the arguments
may be simple identifiers or more complicated expressions.
The apply
element is conceptually necessary in order to distinguish
between a function or operator, and an instance of its use. The expression
constructed by applying a function to 0 or more arguments is always an element from
the codomain of the function. Proper usage depends on the operator that is being
applied. For example, the plus
operator may have zero or more arguments,
while the minus
operator requires one or two arguments in order to be properly
formed.
Strict Content MathML applications are rendered as mathematical
function applications. If
denotes the rendering of
and
the rendering of
, the the sample
rendering of a simple application is as follows:
Content MathML
<apply></apply> |
Sample Presentation
<mrow><mo>⁡</mo> <mrow> <mo fence="true">(</mo> <mo separator="true">,</mo> <mo separator="true">,</mo> <mo separator="true">,</mo> <mo fence="true">)</mo> </mrow> </mrow> |
Non-Strict MathML applications may also be used with qualifiers. In the absence of
any more specific rendering rules for well-known operators, rendering
should follow the sample presentation below, motivated by the typical
presentation for sum
. Let
denote the rendering of
,
the rendering of
, and so on. Then:
Content MathML
<apply><bvar> </bvar> <domainofapplication> </domainofapplication> </apply> |
Sample Presentation
<mrow> <munder><mrow> <mo>∈</mo><!--ELEMENT OF--> </mrow> </munder> <mo>⁡</mo><!--FUNCTION APPLICATION--> <mrow> <mo fence="true">(</mo> <mo fence="true">)</mo> </mrow> </mrow> |
<bind>
and <bvar>
Many complex mathematical expressions are constructed with the use of bound variables, and bound variables are an important concept of logic and formal languages. Variables become bound in the scope of an expression through the use of a quantifier. Informally, they can be thought of as the "dummy variables" in expressions such as integrals, sums, products, and the logical quantifiers "for all" and "there exists". A bound variable is characterized by the property that systematically renaming the variable (to a name not already appearing in the expression) does not change the meaning of the expression.
Schema Fragment (Strict) | Schema Fragment (Full) | |
---|---|---|
Class | Bind | Bind |
Attributes | CommonAtt | CommonAtt, DefEncAtt |
Content | ContExp, BvarQ*, ContExp | ContExp, BvarQ*, Qualifier*, ContExp+ |
Binding expressions are represented as MathML expression trees using the bind
element. Its first child is a MathML expression that represents a binding operator, for
example integral operator. This is followed by a non-empty list of bvar
elements denoting the bound variables, and then the final child which is a general
Content MathML expression, known as the body of the binding.
Schema Fragment (Strict) | Schema Fragment (Full) | |
---|---|---|
Class | BVar | BVar |
Attributes | CommonAtt | CommonAtt, DefEncAtt |
Content | ci | semantics-ci | (ci | semantics-ci), degree? | degree?, (ci | semantics-ci |
The bvar
element is used to denote the bound variable of a binding
expression, e.g. in sums, products, and quantifiers or user defined functions.
The content of a bvar
element is an annotated variable,
i.e. either a content identifier represented by a ci
element or a
semantics
element whose first child is an annotated variable. The
name of an annotated variable of the second kind is the name of its first
child. The name of a bound variable is that of the annotated variable
in the bvar
element.
Bound variables are identified by comparing their names. Such
identification can be made explicit by placing an id
on the ci
element in the bvar
element and referring to it using the xref
attribute on all other instances. An example of this approach is
<bind><csymbol cd="quant1">forall</csymbol> <bvar><ci id="var-x">x</ci></bvar> <apply><csymbol cd="relation1">lt</csymbol> <ci xref="var-x">x</ci> <cn>1</cn> </apply> </bind> |
This id
based approach is especially helpful when constructions
involving bound variables are nested.
It is sometimes necessary to associate additional
information with a bound variable. The information might be
something like a detailed mathematical type, an alternative
presentation or encoding or a domain of application. Such
associations are accomplished in the standard way by replacing
a ci
element (even inside the bvar
element)
by a semantics
element containing both the ci
and the additional information. Recognition of an instance of
the bound variable is still based on the actual ci
elements and not the semantics
elements or anything
else they may contain. The id
based-approach
outlined above may still be used.
The following example encodes forall x. x+y=y+x.
<bind><csymbol cd="quant1">forall</csymbol> <bvar><ci>x</ci></bvar> <apply><csymbol cd="relation1">eq</csymbol> <apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><ci>y</ci></apply> <apply><csymbol cd="arith1">plus</csymbol><ci>y</ci><ci>x</ci></apply> </apply> </bind> |
In non-Strict Content markup, the bvar
element is used in
a number of idiomatic constructs. These are described in Section 4.3.3 Qualifiers and Section 4.4 Content MathML for Specific Operators and Constants.
It is a defining property of bound variables that they can be renamed
consistently in the scope of their parent bind
element.
This operation, sometimes known as α-conversion,
preserves the semantics of the expression.
A bound variable x may be renamed to say y so long as y does not occur free in the body of the binding, or in any annotations of the bound variable, x to be renamed, or later bound variables.
If a bound variable x is renamed, all free occurrences of
x in annotations in its bvar
element,
any following bvar
children of the bind
and in the expression in the body of the bind
should be renamed.
In the example in the previous section, note how renaming x to z produces the equivalent expression forall z. z+y=y+z, whereas x may not be renamed to y, as y is free in the body of the binding and would be captured, producing the expression forall y. y+y=y+y which is not equivalent to the original expression.
If
and
are Content MathML expressions
that render as the Presentation MathML expressions
and
then the sample rendering of a binding element is as follows:
Content MathML
<bind><bvar> </bvar> <bvar> </bvar> <bvar> </bvar> </bind> |
Sample Presentation
<mrow><mrow> <mo separator="true">,</mo> <mo separator="true">,</mo> </mrow> <mo separator="true">.</mo> </mrow> |
<share>
To conserve space in the XML encoding, MathML expression trees can make use of structure sharing.
share
element
Schema Fragment | |
---|---|
Class | Share |
Attributes | CommonAtt, src |
src Attribute Values
|
URI |
Content | Empty |
The share
element has an href
attribute used to
to reference a MathML expression tree. The value of the
href
attribute is a URI specifying the id
attribute of the root node of the expression tree. When building a
MathML expression tree, the share
element is equivalent to a copy of the MathML
expression tree referenced by the href
attribute. Note that this copy is
structurally equal, but not identical to the element referenced. The
values of the share
will often be relative URI references, in which case they
are resolved using the base URI of the document containing the share
element.
For instance, the mathematical object f(f(f(a,a),f(a,a)),f(f(a,a),f(a,a))) can be encoded as either one of the following representations (and some intermediate versions as well).
|
|
Say that an element dominates all its children and all
elements they dominate. Say also that a
share
element dominates its target, i.e. the element that carries the
id
attribute pointed to by the href
attribute. For instance in the
representation on the right above, the apply
element with id="t1"
and also the
second share
(with href="t11"
) both dominate the
apply
element with id="t11"
.
The occurrences of the share
element must obey the following global
acyclicity constraint: An element may not dominate itself. For example, the
following representation violates this constraint:
<apply id="badid1"><csymbol cd="arith1">divide</csymbol> <cn>1</cn> <apply><csymbol cd="arith1">plus</csymbol> <cn>1</cn> <share href="#badid1"/> </apply> </apply>
Here, the apply
element with id="badid1"
dominates its third child,
which dominates the share
element, which dominates its target: the element with
id="badid1"
. So by transitivity, this element dominates itself. By the
acyclicity constraint, the example is not a valid MathML expression tree. It
might be argued that such an expression could be given the interpretation of the continued fraction
.
However, the procedure of building an expression tree by replacing
share
element does not terminate for such an
expression, and hence such expressions are not allowed by Content MathML.
Note that the acyclicity constraints is not restricted to such simple cases, as the following example shows:
<apply id="bar"> <apply id="baz"> <csymbol cd="arith1">plus</csymbol> <csymbol cd="arith1">plus</csymbol> <cn>1</cn> <cn>1</cn> <share href="#baz"/> <share href="#bar"/> </apply> </apply>
Here, the apply
with id="bar"
dominates its third child, the
share
with href="#baz"
. That element dominates its target apply
(with id="baz"
), which in turn dominates its third child, the share
with href="#bar"
. Finally, the share
with
href="#bar"
dominates its target, the original
apply
element with id="bar"
. So this pair of representations
ultimately violates the acyclicity constraint.
Note that the share
element is a syntactic referencing mechanism:
a share
element stands for the exact element it points to. In particular,
referencing does not interact with binding in a semantically intuitive way, since it
allows a phenomenon called variable capture to
occur. Consider an example:
<bind id="outer"><csymbol cd="fns1">lambda</csymbol> <bvar><ci>x</ci></bvar> <apply><ci>f</ci> <bind id="inner"><csymbol cd="fns1">lambda</csymbol> <bvar><ci>x</ci></bvar> <share id="copy" href="#orig"/> </bind> <apply id="orig"><ci>g</ci><ci>x</ci></apply> </apply> </bind> |
This represents a term
which has two sub-terms of the form
,
one with
id="orig"
(the one explicitly represented) and one with id="copy"
,
represented by the share
element.
In the original, explicitly-represented term,
the variable x is bound by the
outer bind
element.
However, in the copy, the variable x is
bound by the inner bind
element.
One says that the inner bind
has captured the variable x.
Using references that capture variables in this way can easily lead to representation
errors, and is not recommended. For instance, using
α-conversion to rename the inner occurrence of x
into, say, y leads to the semantically equivalent expression
.
However, in this form, it is no longer possible to share the expression
.
Replacing x with y in the inner
bvar
without replacing the share
element results in a change
in semantics.
semantics
Content elements can be annotated with additional information via the
semantics
element. MathML uses the
semantics
element to wrap the annotated element and the
annotation-xml
and annotation
elements used for representing the
annotations themselves. The use of the semantics
, annotation
and
annotation-xml
is described in detail Chapter 5 Mixing Markup Languages for Mathematical Expressions.
The semantics
element is be considered part of both
presentation MathML and Content MathML. MathML considers a semantics
element
(strict) Content MathML, if and only if its first child is (strict) Content MathML.
<cerror>
Schema Fragment (Strict) | Schema Fragment (Full) | |
---|---|---|
Class | Error | Error |
Attributes | CommonAtt | CommonAtt, DefEncAtt |
Content | csymbol, ContExp* | csymbol, ContExp* |
A content error expression is made up of a csymbol
followed by a sequence of zero or more MathML expressions. The
initial expression must be a csymbol
indicating the kind of
error. Subsequent children, if present, indicate the context in
which the error occurred.
The cerror
element has no direct mathematical meaning.
Errors occur as the result of some action performed on an expression
tree and are thus of real interest only when some sort of
communication is taking place. Errors may occur inside other objects
and also inside other errors.
As an example, to encode a division by zero error, one might
employ a hypothetical aritherror
Content Dictionary
containing a DivisionByZero
symbol, as in the following
expression:
<cerror> <csymbol cd="aritherror">DivisionByZero</csymbol> <apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn>0</cn></apply> </cerror> |
Note that error markup generally should enclose only the smallest
erroneous sub-expression. Thus a cerror
will often be a sub-expression of
a bigger one, e.g.
<apply><csymbol cd="relation1">eq</csymbol> <cerror> <csymbol cd="aritherror">DivisionByZero</csymbol> <apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn>0</cn></apply> </cerror> <cn>0</cn> </apply> |
The default presentation of a cerror
element is an
merror
expression whose first child is a presentation of the
error symbol, and whose subsequent children are the default
presentations of the remaining children of the cerror
. In
particular, if one of the remaining children of the cerror
is
a presentation MathML expression, it is used literally in the
corresponding merror
.
<cerror> <csymbol cd="aritherror">DivisionByZero</csymbol> <apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn>0</cn></apply> </cerror> |
Sample Presentation
<merror> <mtext>DivisionByZero: </mtext> <mfrac><mi>x</mi><mn>0</mn></mfrac> </merror> |
Note that when the context where an error occurs is so nonsensical that its default presentation would not be useful, an application may provide an alternative representation of the error context. For example:
<cerror> <csymbol cd="error">Illegal bound variable</csymbol> <cs> <bvar><plus/></bvar> </cs> </cerror> |
<cbytes>
Schema Fragment (Strict) | Schema Fragment (Full) | |
---|---|---|
Class | Cbytes | Cbytes |
Attributes | CommonAtt | CommonAtt, DefEncAtt |
Content | base64 | base64 |
The content of cbytes
represents a stream of bytes as a
sequence of characters in Base64 encoding, that is it matches the
base64Binary data type defined in [XMLSchemaDatatypes]. All white space is ignored.
The cbytes
element is mainly used for OpenMath
compatibility, but may be used, as in OpenMath, to encapsulate output
from a system that may be hard to encode in MathML, such as binary
data relating to the internal state of a system, or image data.
The rendering of cbytes
is not expected to represent the
content and the proposed rendering is that of an empty
mrow
. Typically cbytes
is used in an
annotation-xml
or is itself annotated with Presentation
MathML, so this default rendering should rarely be used.
The elements of Strict Content MathML described in the previous section are sufficient to encode logical assertions and expression structure, and they do so in a way that closely models the standard constructions of mathematical logic that underlie the foundations of mathematics. As a consequence, Strict markup can be used to represent all of mathematics, and is ideal for providing consistent mathematical semantics for all Content MathML expressions.
At the same time, many notational idioms of mathematics are not straightforward to represent directly with Strict Content markup. For example, standard notations for sums, integrals, sets, piecewise functions and many other common constructions require non-obvious technical devices, such as the introduction of lambda functions, to rigorously encode them using Strict markup. Consequently, in order to make Content MathML easier to use, a range of additional elements have been provided for encoding such idiomatic constructs more directly. This section discusses the general approach for encoding such idiomatic constructs, and their Strict Content equivalents. Specific constructions are discussed in detail in Section 4.4 Content MathML for Specific Operators and Constants.
Most idiomatic constructions which Content markup addresses fall
into about a dozen classes. Some of these classes, such as container elements, have
their own syntax. Similarly, a small number of non-Strict
constructions involve a single element with an exceptional syntax,
for example partialdiff
. These exceptional elements are
discussed on a case-by-case basis in Section 4.4 Content MathML for Specific Operators and Constants. However, the majority of constructs consist of
classes of operator elements which all share a particular usage of
qualifiers.
These classes of operators are described in Section 4.3.4 Operator Classes.
In all cases, non-Strict expressions may be rewritten using only Strict markup. In most cases, the transformation is completely algorithmic, and may be automated. Rewrite rules for classes of non-Strict constructions are introduced and discussed later in this section, and rewrite rules for exceptional constructs involving a single operator are given in Section 4.4 Content MathML for Specific Operators and Constants. The complete algorithm for rewriting arbitrary Content MathML as Strict Content markup is summarized at the end of the Chapter in Section 4.6 The Strict Content MathML Transformation.
Many mathematical structures are constructed from subparts or
parameters. The motivating example is a set. Informally, one
thinks of a set as a certain kind of mathematical object that
contains a collection of elements. Thus, it is intuitively natural
for the markup for a set to contain, in the XML sense, the markup
for its constituent elements. The markup may define the set
elements explicitly by enumerating them, or implicitly by rule,
using qualifier elements. However, in either case, the markup for
the elements is contained in the markup for the set, and
consequently this style of representation is termed
container markup in MathML. By contrast, Strict
markup represents an instance of a set as the result of applying a
function or constructor symbol to arguments. In this
style of markup, the markup for the set construction is a sibling
of the markup for the set elements in an enclosing apply
element.
While the two approaches are formally equivalent, container markup is generally more intuitive for non-expert authors to use, while Strict markup is preferable is contexts where semantic rigor is paramount. In addition, MathML 2 relied on container markup, and thus container markup is necessary in cases where backward compatibility is required.
MathML provides container markup for the following mathematical
constructs: sets, lists, intervals, vectors, matrices (two
elements), piecewise functions (three elements) and lambda
functions. There are corresponding constructor symbols in Strict
markup for each of these, with the exception of lambda functions,
which correspond to binding symbols in Strict markup. Note that in
MathML 2, the term "container markup" was also taken to include
token elements, and the deprecated declare
, fn
and reln
elements, but MathML 3 limits usage of the term
to the above constructs.
The rewrite rules for obtaining equivalent Strict Content markup from container markup depend on the operator class of the particular operator involved. For details about a specific container element, obtain its operator class (and any applicable special case information) by consulting the syntax table and discussion for that element in Section 4.4 Content MathML for Specific Operators and Constants. Then apply the rewrite rules for that specific operator class as described in Section 4.3.4 Operator Classes.
The arguments to container elements corresponding to
constructors may either be explicitly given as a sequence of child
elements, or they may be specified by a rule using qualifiers. The
only exceptions are the piecewise
, piece
, and
otherwise
elements used for representing functions with
piecewise definitions. The
arguments of these elements must always be specified
explicitly.
Here is an example of container markup with explicitly specified arguments:
<set><ci>a</ci><ci>b</ci><ci>c</ci></set> |
This is equivalent to the following Strict Content MathML expression:
<apply><csymbol cd="set1">set</csymbol><ci>a</ci><ci>b</ci><ci>c</ci></apply> |
Another example of container markup, where the list of arguments is given indirectly as an expression with a bound variable. The container markup for the set of even integers is:
<set> <bvar><ci>x</ci></bvar> <domainofapplication><integers/></domainofapplication> <apply><times/><cn>2</cn><ci>x</ci></apply> </set> |
This may be written as follows in Strict Content MathML:
<apply><csymbol cd="set1">map</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>x</ci></bvar> <apply><csymbol cd="arith1">times</csymbol> <cn>2</cn> <ci>x</ci> </apply> </bind> <csymbol cd="setname1">Z</csymbol> </apply> |
The lambda
element is a container element
corresponding to the lambda symbol
in the fns1 Content Dictionary. However, unlike the
container elements of the preceding section, which purely
construct mathematical objects from arguments, the lambda
element performs variable binding as well. Therefore, the child
elements of lambda
have distinguished roles. In
particular, a lambda
element must have at least one
bvar
child, optionally followed by qualifier elements, followed by a
Content MathML element. This basic difference between the
lambda
container and the other constructor container
elements is also reflected in the OpenMath symbols to which they
correspond. The constructor symbols have an OpenMath role of
"application", while the lambda symbol has a role of "bind".
This example shows the use of lambda
container element and the equivalent use of bind
in Strict Content MathML
<lambda><bvar><ci>x</ci></bvar><ci>x</ci></lambda> |
<bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>x</ci></bvar><ci>x</ci> </bind> |
<apply>
MathML allows the use of the apply
element to perform
variable binding in non-Strict constructions instead of
the bind
element. This usage conserves backwards
compatibility with MathML 2. It also simplifies the encoding of
several constructs involving bound variables with qualifiers as
described below.
Use of the apply
element to bind variables is allowed
in two situations. First, when the operator to be applied is
itself a binding operator, the apply
element merely
substitutes for the bind
element. The logical quantifiers
<forall/>
, <exists/>
and the
container element lambda
are the primary examples of this
type.
The second situation arises when the operator being applied allows the use of bound variables with qualifiers. The most common examples are sums and integrals. In most of these cases, the variable binding is to some extent implicit in the notation, and the equivalent Strict representation requires the introduction of auxiliary constructs such as lambda expressions for formal correctness.
Because expressions using bound variables with qualifiers are
idiomatic in nature, and do not always involve true variable
binding, one cannot expect systematic renaming (alpha-conversion)
of variables "bound" with apply
to preserve meaning in
all cases. An example for this is the diff
element where
the bvar
term is technically not bound at all.
The following example illustrates the use of apply
with a binding operator. In these cases, the corresponding Strict
equivalent merely replaces the apply
element with a
bind
element:
<apply><forall/> <bvar><ci>x</ci></bvar> <apply><geq/><ci>x</ci><ci>x</ci></apply> </apply> |
The equivalent Strict expression is:
<bind><csymbol cd="logic1">forall</csymbol> <bvar><ci>x</ci></bvar> <apply><csymbol cd="relation1">geq</csymbol><ci>x</ci><ci>x</ci></apply> </bind> |
In this example, the sum operator is not itself a binding operator, but bound variables with qualifiers are implicit in the standard notation, which is reflected in the non-Strict markup. In the equivalent Strict representation, it is necessary to convert the summand into a lambda expression, and recast the qualifiers as an argument expression:
<apply><sum/> <bvar><ci>i</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><cn>100</cn></uplimit> <apply><power/><ci>x</ci><ci>i</ci></apply> </apply> |
The equivalent Strict expression is:
<apply><csymbol cd="arith1">sum</csymbol> <apply><csymbol cd="interval1">integer_interval</csymbol> <cn>0</cn> <cn>100</cn> </apply> <bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>i</ci></bvar> <apply><csymbol cd="arith1">power</csymbol> <ci>x</ci> <ci>i</ci> </apply> </bind> </apply> |
Many common mathematical constructs involve an operator together with some additional data. The additional data is either implicit in conventional notation, such as a bound variable, or thought of as part of the operator, as is the case with the limits of a definite integral. MathML 3 uses qualifier elements to represent the additional data in such cases.
Qualifier elements are always used in conjunction with operator or container
elements. Their meaning is idiomatic, and depends on the context in which they are
used. When used with an operator, qualifiers always follow the operator and precede
any arguments that are present. In all cases, if more than one qualifier is present,
they appear in the order bvar
, lowlimit
, uplimit
,
interval
, condition
, domainofapplication
, degree
,
momentabout
, logbase
.
The precise function of qualifier elements depends on the operator or container that they modify. The majority of use cases fall into one of several categories, discussed below, and usage notes for specific operators and qualifiers are given in Section 4.4 Content MathML for Specific Operators and Constants.
<domainofapplication>
,
<interval>
,
<condition>
,
<lowlimit>
and
<uplimit>
Class | qualifier |
---|---|
Attributes | CommonAtt |
Content | ContExp |
(For the syntax of interval
see Section 4.4.1.1 Interval <interval>
.)
The primary use of domainofapplication
, interval
,
uplimit
, lowlimit
and condition
is to
restrict the values of a bound variable. The most general qualifier
is domainofapplication
. It is used to specify a set (perhaps
with additional structure, such as an ordering or metric) over which
an operation is to take place. The interval
qualifier, and
the pair lowlimit
and uplimit
also restrict a bound
variable to a set in the special case where the set is an
interval. The condition
qualifier, like
domainofapplication
, is general, and can be used to restrict
bound variables to arbitrary sets. However, unlike the other
qualifiers, it restricts the bound variable by specifying a
Boolean-valued function of the bound variable. Thus,
condition
qualifiers always contain instances of the bound
variable, and thus require a preceding bvar
, while the other
qualifiers do not. The other qualifiers may even be used when no
variables are being bound, e.g. to indicate the restriction of a
function to a subdomain.
In most cases, any of the qualifiers capable of representing the
domain of interest can be used interchangeably. The most general
qualifier is domainofapplication
, and therefore has a
privileged role. It is the preferred form, unless there are
particular idiomatic reasons to use one of the other qualifiers,
e.g. limits for an integral. In MathML 3, the other forms are treated
as shorthand notations for domainofapplication
because they
may all be rewritten as equivalent domainofapplication
constructions. The rewrite rules to do this are given below. The other
qualifier elements are provided because they correspond to common
notations and map more easily to familiar presentations. Therefore,
in the situations where they naturally arise, they may be more
convenient and direct than domainofapplication
.
To illustrate these ideas, consider the following examples showing alternative
representations of a definite integral. Let C denote the interval from 0 to 1,
and f(x) = x2. Then
domainofapplication
could be used express the integral of a
function f over
C in this way:
<apply><int/> <domainofapplication> <ci type="set">C</ci> </domainofapplication> <ci type="function">f</ci> </apply> |
Note that no explicit bound variable is identified in this
encoding, and the integrand is a function. Alternatively, the
interval
qualifier could be used with an explicit bound variable:
<apply><int/> <bvar><ci>x</ci></bvar> <interval><cn>0</cn><cn>1</cn></interval> <apply><power/><ci>x</ci><cn>2</cn></apply> </apply> |
The pair lowlimit
and uplimit
can also be used.
This is perhaps the most "standard" representation of this integral:
<apply><int/> <bvar><ci>x</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><cn>1</cn></uplimit> <apply><power/><ci>x</ci><cn>2</cn></apply> </apply> |
Finally, here is the same integral, represented using
a condition
on the bound variable:
<apply><int/> <bvar><ci>x</ci></bvar> <condition> <apply><and/> <apply><leq/><cn>0</cn><ci>x</ci></apply> <apply><leq/><ci>x</ci><cn>1</cn></apply> </apply> </condition> <apply><power/><ci>x</ci><cn>2</cn></apply> </apply> |
Note the use of the explicit bound variable within the
condition
term. Note also that when a bound
variable is used, the integrand is an expression in the bound
variable, not a function.
The general technique of using a condition
element
together with domainofapplication
is quite powerful. For
example, to extend the previous example to a multivariate domain, one
may use an extra bound variable and a domain of application
corresponding to a cartesian product:
<apply><int/> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <domainofapplication> <set> <bvar><ci>t</ci></bvar> <bvar><ci>u</ci></bvar> <condition> <apply><and/> <apply><leq/><cn>0</cn><ci>t</ci></apply> <apply><leq/><ci>t</ci><cn>1</cn></apply> <apply><leq/><cn>0</cn><ci>u</ci></apply> <apply><leq/><ci>u</ci><cn>1</cn></apply> </apply> </condition> <list><ci>t</ci><ci>u</ci></list> </set> </domainofapplication> <apply><times/> <apply><power/><ci>x</ci><cn>2</cn></apply> <apply><power/><ci>y</ci><cn>3</cn></apply> </apply> </apply> |
Note that the order of the inner and outer bound variables is significant.
Mappings to Strict Content MathML
When rewriting expressions to Strict Content MathML, qualifier
elements are removed via a series of rules described in this section.
The general algorithm for rewriting a MathML expression involving
qualifiers proceeds in two steps. First, constructs using the
interval
, condition
, uplimit
and
lowlimit
qualifiers are converted to constructs using only
domainofapplication
. Second, domainofapplication
expressions are then rewritten as Strict Content markup.
<apply><bvar> </bvar> <lowlimit> </lowlimit> <uplimit> </uplimit> </apply> |
<apply><bvar> </bvar> <domainofapplication> <apply><csymbol cd="interval1"> </csymbol> </apply> </domainofapplication> </apply> |
The symbol used in this translation depends on the head of the
application, denoted by
here. By default interval should be
used, unless the semantics of the head term can be determined and
indicate a more specific interval symbols. In particular, several
predefined Content MathML element should be used with more specific
interval symbols. If the head is
int
then oriented_interval is used. When the head term
is sum
or product
, integer_interval should be used.
The above technique for replacing lowlimit
and uplimit
qualifiers
with a domainofapplication
element is also used for replacing the
interval
qualifier.
The condition
qualifier restricts a bound variable by specifying a
Boolean-valued expression on a larger domain, specifying whether a given value is in the
restricted domain. The condition
element contains a single child that represents
the truth condition. Compound conditions are formed by applying Boolean operators such as
and
in the condition.
To rewrite an expression using the condition
qualifier as one using domainofapplication
,
<bvar></bvar> <bvar> </bvar> <condition> </condition>
is rewritten to
<bvar></bvar> <bvar> </bvar> <domainofapplication> <apply><csymbol cd="set1">suchthat</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> <bvar> </bvar> </bind> </apply> </domainofapplication>
If the apply
has a domainofapplication
(perhaps originally expressed as
interval
or an uplimit
/lowlimit
pair) then that is used for
. Otherwise
is a set determined by the
type
attribute
of the bound variable as specified in Section 4.2.2.2 Non-Strict uses of <ci>
, if that is
present. If the type is unspecified, the translation introduces an unspecified domain via
content identifier <ci>R</ci>
.
By applying the rules above, expression using the
interval
, condition
, uplimit
and
lowlimit
can be rewritten using only
domainofapplication
. Once a domainofapplication
has
been obtained, the final mapping to Strict markup is accomplished
using the following rules:
An application of a function that is qualified by the
domainofapplication
qualifier (expressed by an apply
element without
bound variables) is converted to an application of a function term constructed with the
restriction symbol.
<apply><domainofapplication> </domainofapplication> </apply> |
may be written as:
<apply> <apply><csymbol cd="fns1">restriction</csymbol></apply> </apply> |
In general, an application involving bound variables and (possibly)
domainofapplication
is rewritten using the following rule,
which makes the domain the first positional argument of the application,
and uses the lambda symbol to encode the variable bindings.
Certain classes of operator have alternative rules, as described below.
A content MathML expression with bound variables and
domainofapplication
<apply><bvar> </bvar> ... <bvar> </bvar> <domainofapplication> </domainofapplication> ... </apply>
is rewritten to
<apply><bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> ... <bvar> </bvar> </bind> ... <bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> ... <bvar> </bvar> </bind> </apply>
If there is no domainofapplication
qualifier the child is
omitted.
<degree>
Class | qualifier |
---|---|
Attributes | CommonAtt |
Content | ContExp |
The degree
element is a qualifier used to specify the
"degree" or "order" of an operation. MathML uses the
degree
element in this way in three contexts: to specify the degree of a
root, a moment, and in various derivatives. Rather than introduce special elements for
each of these families, MathML provides a single general construct, the
degree
element in all three cases.
Note that the degree
qualifier is not used to restrict a bound variable in
the same sense of the qualifiers discussed above. Indeed, with roots and moments, no
bound variable is involved at all, either explicitly or implicitly. In the case of
differentiation, the degree
element is used in conjunction with a
bvar
, but even in these cases, the variable may not be genuinely bound.
For the usage of degree
with the root
and moment
operators, see the discussion of those
operators below. The usage of degree
in differentiation is more complex. In
general, the degree
element indicates the order of the derivative with
respect to that variable. The degree element is allowed as the second child of a
bvar
element identifying a variable with respect to which the derivative is
being taken. Here is an example of a second derivative using the degree
qualifier:
<apply><diff/> <bvar> <ci>x</ci> <degree><cn>2</cn></degree> </bvar> <apply><power/><ci>x</ci><cn>4</cn></apply> </apply> |
For details see Section 4.4.4.2 Differentiation <diff/>
and Section 4.4.4.3 Partial Differentiation <partialdiff/>
.
The Content MathML elements described in detail in the next section may be broadly separated into classes. The class of each element is shown in the syntax table that introduces the element in Section 4.4 Content MathML for Specific Operators and Constants. The class gives an indication of the general intended mathematical usage of the element, and also determines its usage as determined by the schema. The class also determines the applicable rewrite rules for mapping to Strict Content MathML. This section presents the rewrite rules for each of the operator classes.
The rules in this section cover the use cases applicable to
specific operator classes. Special-case rewrite rules for individual
elements are discussed in the sections below. However, the most
common usage pattern is generic, and is used by operators from almost all
operator classes. It consists of applying an operator to an explicit list
of arguments using an apply
element. In these cases,
rewriting to Strict Content MathML is simply a matter of replacing the
empty element with an appropriate csymbol
, as listed in the
syntax tables in Section 4.4 Content MathML for Specific Operators and Constants. This is summarized in
the following rule.
In MathML 2, the definitionURL
attribute could be
used to redefine or modify the meaning of an operator element. When the definitionURL
attribute is present, the value for the cd
attribute on the csymbol
should be
determined by the definitionURL
value if possible. The correspondence between cd
and definitionURL
values is described Section 4.2.3.2 Non-Strict uses of <csymbol>
.
Many MathML operators may be used with an arbitrary number of
arguments. The corresponding OpenMath symbols for elements in these classes
also take an arbitrary number of arguments.
In all such cases, either the arguments my be given
explicitly as children of the apply
or bind
element, or
the list may be specified implicitly via the use of qualifier
elements.
The elements representing these n-ary operators are specified in the following schema patterns in Appendix A Parsing MathML: nary-arith.class, nary-functional.class, nary-logical.class, nary-linalg.class, nary-set.class, nary-constructor.class.
If the argument list is given explicitly, the Rewrite: element rule applies.
Any use of qualifier elements is expressed in Strict Content
MathML, via explicitly applying the function to a list of arguments
using the apply_to_list symbol as shown
in the following rule. The rule only considers the
domainofapplication
qualifier as other qualifiers may be
rewritten to domainofapplication
as described earlier.
An expression of the following form,
where represents any
element of the relevant class and
is an arbitrary expression involving the bound variable(s)
<apply><bvar> </bvar> <domainofapplication> </domainofapplication> </apply> |
is rewritten to
<apply><csymbol cd="fns2">apply_to_list</csymbol> <csymbol cd=""> </csymbol> <apply><csymbol cd="list1">map</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> </bind> </apply> </apply> |
The above rule applies to all symbols in the listed classes.
In the case of nary-set.class the choice of Content
Dictionary to use depends on the type
attribute on the
arguments, defaulting to set1, but multiset1
should be used if type
="multiset".
Note that the members of the nary-constructor.class, such
as vector
, use constructor syntax where the arguments and
qualifiers are given as children of the element rather than as
children of a containing apply
. In this case, the above rules apply
with the analogous syntactic modifications.
The use of set
and list
follows the same format
as other n-ary constructors, however when rewriting to Strict
Content MathML a variant of the above rule is used. This is because the map
symbol implicitly constructs the required set or list, and apply_to_list is
not needed in this case.
The elements representing these n-ary operators are specified in the schema pattern nary-setlist-constructor.class.
If the argument list is given explicitly, the Rewrite: element rule applies.
When qualifiers are used to specify the list of arguments, the following rule is used.
An expression of the following form,
where is either of the elements
set
or list
and
is an arbitrary expression involving the bound variable(s)
<set> <bvar></bvar> <domainofapplication> </domainofapplication> </set> |
is rewritten to
<apply><csymbol cd="">map</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> </bind> </apply> |
Note that
when is already a set
or list of the appropriate type for the container element, and the lambda function created
from
is
the identity, the entire container element should be rewritten
directly as
.
In the case of set
, the choice of Content
Dictionary and symbol depends on the value of the type
attribute of the arguments. By default the set symbol is used, but if one of the arguments has
type
attribute with value "multiset", the multiset symbol is used.
If there is a type
attribute with value other than "set" or "multiset"
the set symbol should be used, and the arguments should be annotated with their type
by rewriting the type
attribute using the rule
Rewrite: attributes.
MathML allows transitive relations to be used with multiple arguments, to give a natural expression to "chains" of relations such as a < b < c < d. However unlike the case of the arithmetic operators, the underlying symbols used in the Strict Content MathML are classed as binary, so it is not possible to use apply_to_list as in the previous section, but instead a similar function predicate_on_list is used, the semantics of which is essentially to take the conjunction of applying the predicate to elements of the domain two at a time.
The elements representing these n-ary operators are specified in the following schema patterns in Appendix A Parsing MathML: nary-reln.class, nary-set-reln.class.
An expression of the form
<apply></apply> |
rewrites to Strict Content MathML
<apply><csymbol cd="fns2">predicate_on_list</csymbol> <csymbol cd=""> </csymbol> <apply><csymbol cd="list1">list</csymbol> </apply> </apply> |
An expression of the form
<apply><bvar> </bvar> <domainofapplication> </domainofapplication> </apply> |
where
is an arbitrary expression involving the bound variable, rewrites to the Strict Content MathML
<apply><csymbol cd="fns2">predicate_on_list</csymbol> <csymbol cd=""> </csymbol> <apply><csymbol cd="list1">map</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> </bind> </apply> </apply> |
The above rules apply to all symbols in classes nary-reln.class
and nary-set-reln.class. In the latter case the choice of Content
Dictionary to use depends on the type
attribute on the
symbol, defaulting to set1, but multiset1
should be used if type
="multiset".
The MathML elements, max
, min
and some statistical
elements such as mean
may be used as a n-ary function as in
the above classes, however a special interpretation is given in the
case that a single argument is supplied. If a single argument is
supplied the function is applied to the elements represented by the
argument.
The underlying symbol used in Strict Content MathML for these elements is Unary and so if the MathML is used with 0 or more than 1 arguments, the function is applied to the set constructed from the explicitly supplied arguments according to the following rule.
The elements representing these n-ary operators are specified in the following schema patterns in Appendix A Parsing MathML: nary-minmax.class, nary-stats.class.
When an element,
, of class nary-stats or nary-minmax
is applied to an explicit
list of 0 or 2 or more arguments,
<apply></apply> |
It is is translated to the unary application of the symbol
<csymbol cd="
as specified in the syntax table for the element to the set of
arguments, constructed using the
" name=" "/><csymbol cd="set1" name="set"/>
symbol.
<apply><csymbol cd=""> </csymbol> <apply><csymbol cd="set1">set</csymbol> </apply> </apply> |
Like all MathML n-ary operators, The list of arguments may be specified implicitly using qualifier elements. This is expressed in Strict Content MathML using the following rule, which is similar to the rule Rewrite: n-ary domainofapplication but differs in that the symbol can be directly applied to the constructed set of arguments and it is not necessary to use apply_to_list.
An expression of the following form,
where represents any
element of the relevant class and
is an arbitrary expression involving the bound variable(s)
<apply><bvar> </bvar> <domainofapplication> </domainofapplication> </apply> |
is rewritten to
<apply><csymbol cd=""> </csymbol> <apply><csymbol cd="set1">map</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> </bind> </apply> </apply> |
Note that
when is already a set
and the lambda function created from
is
the identity, the
domainofapplication
term should should be
rewritten directly
as .
If the element is applied to a single argument the set symbol is not used and the symbol is applied directly to the argument.
When an element,
, of class nary-stats or nary-minmax
is applied to a single argument,
<apply></apply> |
It is is translated to the unary application of the symbol in the syntax table for the element.
<apply><csymbol cd=""> </csymbol> </apply> |
Note: Earlier versions of MathML were not explicit about the correct interpretation of elements in this class, and left it undefined as to whether an expression such as max(X) was a trivial application of max to a singleton, or whether it should be interpreted as meaning the maximum of values of the set X. Applications finding that the rule Rewrite: n-ary unary single can not be applied as the supplied argument is a scalar may wish to use the rule Rewrite: n-ary unary set as an error recovery. As a further complication, in the case of the statistical functions the Content Dictionary to use in this case depends on the desired interpretation of the argument as a set of explicit data or a random variable representing a distribution.
Binary operators take two arguments and simply map to OpenMath
symbols via Rewrite: element
without the need of any special rewrite rules. The binary
constructor interval
is similar but uses constructor syntax
in which the arguments are children of the element, and the symbol
used depends on the type element as described in Section 4.4.1.1 Interval <interval>
The elements representing these binary operators are specified in the following schema patterns in Appendix A Parsing MathML: binary-arith.class, binary-logical.class, binary-reln.class, binary-linalg.class, binary-set.class.
Unary operators take a single argument and map to OpenMath symbols via Rewrite: element without the need of any special rewrite rules.
The elements representing these unary operators are specified in the following schema patterns in Appendix A Parsing MathML: unary-arith.class, unary-functional.class, unary-set.class, unary-elementary.class, unary-veccalc.class.
Constant symbols relate to mathematical constants such as e and true and also to names of sets such as the Real Numbers, and Integers. In Strict Content MathML, they rewrite simply to the corresponding symbol listed in the syntax tables for these elements in Section 4.4.10 Constant and Symbol Elements.
The elements representing these constants are specified in the schema patterns constant-arith.class and constant-set.class.
The Quantifier class is used for the forall and exists quantifiers of predicate calculus.
The elements representing quantifiers are specified in the schema pattern quantifier.class.
If used with bind
and no qualifiers,
then the interpretation in Strict Content MathML is simple. In general
if used with apply
or qualifiers, the interpretation in
Strict Content MathML is via the following rule.
An expression of following form where
denotes an element of
class quantifier and
is an arbitrary expression involving the bound variable(s)
<apply><bvar> </bvar> <domainofapplication> </domainofapplication> </apply> |
is rewritten to an expression
<bind><csymbol cd=""> </csymbol> <bvar> </bvar> <apply><csymbol cd=" "> </csymbol> <apply><csymbol cd="set1">in</csymbol> </apply> </apply> </bind> |
where the symbols
<csymbol cd="
and
"> </csymbol><csymbol cd="
are as specified in the syntax table of the element.
(The additional symbol being
"> </csymbol>and in the case of exists
and
implies in the case of forall
.) When no
domainofapplication
is present, no logical conjunction is necessary, and the translation
is direct.
Special purpose classes, described in the sections for the appropriate elements
The elements are specified in the following schema patterns in Appendix A Parsing MathML: lambda.class, interval.class, int.class, partialdiff.class, sum.class, product.class, limit.class.
A number of content MathML elements such as cn
and
interval
allow attributes to specialize the semantics of the
objects they represent. For these cases, special rewrite rules are
given on a case-by-case basis in Section 4.4 Content MathML for Specific Operators and Constants. However,
content MathML elements also accept attributes shared all MathML elements, and
depending on the context, may also contain attributes from other XML
namespaces. Such attributes must be rewritten in alternative form in
Strict Content Markup.
For instance,
<ci class="" xmlns:other=" " other:att=" "> </ci> |
is rewritten to
<semantics> <ci></ci> <annotation cd="mathmlattr" name="class" encoding="text/plain"> </annotation> <annotation-xml cd="mathmlattr" name="foreign" encoding="MathML-Content"> <apply><csymbol cd="mathmlattr">foreign_attribute</csymbol> <cs> </cs> <cs> </cs> <cs> </cs> <cs> </cs> </apply> </annotation-xml> </semantics> |
For MathML attributes not allowed in Strict Content MathML the content dictionary mathmlattr is referenced, which provides symbols for all attributes allowed on content MathML elements.
This section presents elements representing a core set of mathematical operators, functions and constants. Most are empty elements, covering the subject matter of standard mathematics curricula up to the level of calculus. The remaining elements are container elements for sets, intervals, vectors and so on. For brevity, all elements defined in this section are sometimes called operator elements.
Each subsection below discusses a specific operator element, beginning with a syntax table, giving the elements operator class. Special case rules for rewriting as Strict Markup are introduced as needed. However, in most cases, the generic rewrite rules for the appropriate operator class is sufficient. In particular, unless otherwise indicated, elements are to be rewritten using the default Rewrite: element rule. Note, however, that all elements in this section must be rewritten in some fashion, since they are not allowed in Strict Content markup.
In MathML 2, the definitionURL
attribute could be
used to redefine or modify the meaning of an operator
element. This use of the definitionURL
attribute is deprecated in MathML 3. Instead a
csymbol
element should be used. In general, the value of
cd
attribute on the csymbol
will correspond to
the definitionURL
value.
<interval>
Class | interval |
---|---|
Attributes | CommonAtt, DefEncAtt,closure? |
Content | ContExp,ContExp |
OM Symbols | interval_cc, interval_oc, interval_co, interval_oo |
The interval
element is a container element used to represent simple mathematical intervals of the
real number line. It takes an optional attribute closure
, with a default value
of "closed".
Content MathML
<interval closure="open"><ci>x</ci><cn>1</cn></interval> |
<interval closure="closed"><cn>0</cn><cn>1</cn></interval> |
<interval closure="open-closed"><cn>0</cn><cn>1</cn></interval> |
<interval closure="closed-open"><cn>0</cn><cn>1</cn></interval> |
Sample Presentation
<mfenced><mi>x</mi><mn>1</mn></mfenced> |
<mfenced open="[" close="]"><mn>0</mn><mn>1</mn></mfenced> |
<mfenced open="(" close="]"><mn>0</mn><mn>1</mn></mfenced> |
<mfenced open="[" close=")"><mn>0</mn><mn>1</mn></mfenced> |
Mapping to Strict Content MathML
In Strict markup, the interval
element corresponds to one
of four symbols from the interval1 content
dictionary. If closure
has the value "open" then
interval
corresponds to the
interval_oo.
With the value "closed"
interval
corresponds to the symbol
interval_cc,
with value "open-closed" to
interval_oc, and with
"closed-open" to
interval_co.
<inverse>
Class | unary-functional |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | inverse |
The inverse
element is applied to a function in order to
construct a generic expression for the functional inverse of that
function. The inverse
element may either be applied to
arguments, or it may appear alone, in which case it represents an
abstract inversion operator acting on other functions.
Content MathML
<apply><inverse/> <ci> f </ci> </apply> |
Sample Presentation
<msup><mi>f</mi><mrow><mo>(</mo><mn>-1</mn><mo>)</mo></mrow></msup> |
Content MathML
<apply> <apply><inverse/><ci type="matrix">A</ci></apply> <ci>a</ci> </apply> |
Sample Presentation
<mrow> <msup><mi>A</mi><mrow><mo>(</mo><mn>-1</mn><mo>)</mo></mrow></msup> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>a</mi></mfenced> </mrow> |
<lambda>
Class | lambda |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | BvarQ, DomainQ, ContExp |
Qualifiers | BvarQ,DomainQ |
OM Symbols | lambda |
The lambda
element is used to construct a user-defined
function from an expression, bound variables, and qualifiers. In a
lambda construct with n (possibly 0) bound variables, the
first n children are bvar
elements that identify
the variables that are used as placeholders in the last child for
actual parameter values. The bound variables can be restricted by an
optional domainofapplication
qualifier or one of its
shorthand
notations. The meaning of the lambda
construct is an
n-ary function that returns the expression in the last
child where the bound variables are replaced with the respective
arguments.
The domainofapplication
child restricts the possible
values of the arguments of the constructed function. For instance, the
following lambda
construct represents a function on
the integers.
<lambda> <bvar><ci> x </ci></bvar> <domainofapplication><integers/></domainofapplication> <apply><sin/><ci> x </ci></apply> </lambda> |
If a lambda
construct does not contain bound variables, then
the lambda
construct is superfluous and may be removed,
unless it also contains a domainofapplication
construct.
In that case, if the last child of the lambda
construct is
itself a function, then the domainofapplication
restricts
its existing functional arguments, as in this example, which is
a variant representation for the function above.
<lambda> <domainofapplication><integers/></domainofapplication> <sin/> </lambda> |
Otherwise, if the last child of the lambda
construct is not a
function, say a number, then the lambda
construct will not be
a function, but the same number, and any domainofapplication
is ignored.
Content MathML
<lambda> <bvar><ci>x</ci></bvar> <apply><sin/> <apply><plus/><ci>x</ci><cn>1</cn></apply> </apply> </lambda> |
Sample Presentation
<mrow> <mi>λ<!--GREEK SMALL LETTER LAMDA--></mi> <mi>x</mi> <mo>.</mo> <mfenced> <mrow> <mi>sin</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow> </mrow> </mfenced> </mrow> |
<mrow> <mi>x</mi> <mo>↦<!--RIGHTWARDS ARROW FROM BAR--></mo> <mrow> <mi>sin</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow> </mrow> </mrow> |
Mapping to Strict Markup
If the lambda
element does not contain qualifiers, the
lambda expression is directly translated into a bind
expression.
<lambda> <bvar></bvar><bvar> </bvar> </lambda> |
rewrites to the Strict Content MathML
<bind><csymbol cd="fns1">lambda</csymbol> <bvar></bvar><bvar> </bvar> </bind> |
If the lambda
element does contain qualifiers, the
qualifier may be rewritten to domainofapplication
and then the lambda expression is translated to a
function term constructed with lambda
and restricted to the specified domain using
restriction.
<lambda> <bvar></bvar><bvar> </bvar> <domainofapplication> </domainofapplication> </lambda> |
rewrites to the Strict Content MathML
<apply><csymbol cd="fns1">restriction</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar></bvar><bvar> </bvar> </bind> </apply> |
<compose/>
Class | nary-functional |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | left_compose |
The compose
element represents the function
composition operator. Note that MathML makes no assumption about the domain
and codomain of the constituent functions in a composition; the domain of the
resulting composition may be empty.
The compose
element is a commutative n-ary operator. Consequently, it may be
lifted to the induced operator defined on a collection of arguments indexed by a (possibly
infinite) set by using qualifier elements as described in Section 4.3.4.1 N-ary Operators (classes nary-arith, nary-functional, nary-logical,
nary-linalg, nary-set, nary-constructor).
Content MathML
<apply><compose/><ci>f</ci><ci>g</ci><ci>h</ci></apply> |
Sample Presentation
<mrow><mi>f</mi><mo>∘<!--RING OPERATOR--></mo><mi>g</mi><mo>∘<!--RING OPERATOR--></mo><mi>h</mi></mrow> |
Content MathML
<apply><eq/> <apply> <apply><compose/><ci>f</ci><ci>g</ci></apply> <ci>x</ci> </apply> <apply><ci>f</ci><apply><ci>g</ci><ci>x</ci></apply></apply> </apply> |
Sample Presentation
<mrow> <mrow> <mrow><mo>(</mo><mi>f</mi><mo>∘<!--RING OPERATOR--></mo><mi>g</mi><mo>)</mo></mrow> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>x</mi></mfenced> </mrow> <mo>=</mo> <mrow> <mi>f</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced> <mrow> <mi>g</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>x</mi></mfenced> </mrow> </mfenced> </mrow> </mrow> |
<ident/>
Class | unary-functional |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | identity |
The ident
element represents the
identity function. Note that MathML makes no assumption about the
domain and codomain of the represented identity function, which
depends on the context in which it is used.
Content MathML
<apply><eq/> <apply><compose/> <ci type="function">f</ci> <apply><inverse/> <ci type="function">f</ci> </apply> </apply> <ident/> </apply> |
Sample Presentation
<mrow> <mrow> <mi>f</mi> <mo>∘<!--RING OPERATOR--></mo> <msup><mi>f</mi><mrow><mo>(</mo><mn>-1</mn><mo>)</mo></mrow></msup> </mrow> <mo>=</mo> <mi>id</mi> </mrow> |
<domain/>
Class | unary-functional |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | domain |
The domain
element represents the domain of the
function to which it is applied. The domain is the set of values
over which the function is defined.
Content MathML
<apply><eq/> <apply><domain/><ci>f</ci></apply> <reals/> </apply> |
Sample Presentation
<mrow> <mrow><mi>domain</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>f</mi></mfenced></mrow> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> |
<codomain/>
Class | unary-functional |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | range |
The codomain
represents the codomain, or range, of the function
to which is is applied. Note that the codomain is not necessarily
equal to the image of the function, it is merely required to contain
the image.
Content MathML
<apply><eq/> <apply><codomain/><ci>f</ci></apply> <rationals/> </apply> |
Sample Presentation
<mrow> <mrow><mi>codomain</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>f</mi></mfenced></mrow> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> </mrow> |
<image/>
Class | unary-functional |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | image |
The image
element represent the image of
the function to which it is applied. The image of a function is the
set of values taken by the function. Every point in the image is
generated by the function applied to some point of the domain.
Content MathML
<apply><eq/> <apply><image/><sin/></apply> <interval><cn>-1</cn><cn> 1</cn></interval> </apply> |
Sample Presentation
<mrow> <mrow><mi>image</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>sin</mi></mfenced></mrow> <mo>=</mo> <mfenced open="[" close="]"><mn>-1</mn><mn>1</mn></mfenced> </mrow> |
<piecewise>
, <piece>
, <otherwise>
piecewise
|
piece
|
otherwise
|
The piecewise
, piece
, and otherwise
elements are used to
represent "piecewise" function definitions of the form "
H(x) = 0 if x less than 0, H(x) = 1
otherwise".
The declaration is constructed using the piecewise
element. This contains
zero or more piece
elements, and optionally one otherwise
element. Each
piece
element contains exactly two children. The first child defines the value
taken by the piecewise
expression when the condition specified in the associated
second child of the piece
is true. The degenerate case of no piece
elements and no otherwise
element is treated as undefined for all values of the
domain.
The otherwise
element allows the specification of a value to be taken by the
piecewise
function when none of the conditions (second child elements of the
piece
elements) is true, i.e. a default value.
It should be noted that no "order of execution" is implied by the
ordering of the piece
child elements within piecewise
. It is the
responsibility of the author to ensure that the subsets of the function domain defined by
the second children of the piece
elements are disjoint, or that, where they
overlap, the values of the corresponding first children of the piece
elements
coincide. If this is not the case, the meaning of the expression is
undefined.
Here is an example:
Content MathML
<piecewise> <piece> <apply><minus/><ci>x</ci></apply> <apply><lt/><ci>x</ci><cn>0</cn></apply> </piece> <piece> <cn>0</cn> <apply><eq/><ci>x</ci><cn>0</cn></apply> </piece> <piece> <ci>x</ci> <apply><gt/><ci>x</ci><cn>0</cn></apply> </piece> </piecewise> |
Sample Presentation
<mrow> <mo>{</mo> <mtable> <mtr> <mtd><mrow><mo>−<!--MINUS SIGN--></mo><mi>x</mi></mrow></mtd> <mtd columnalign="left"><mtext> <!--NO-BREAK SPACE--> if  <!--NO-BREAK SPACE--></mtext></mtd> <mtd><mrow><mi>x</mi><mo><</mo><mn>0</mn></mrow></mtd> </mtr> <mtr> <mtd><mn>0</mn></mtd> <mtd columnalign="left"><mtext> <!--NO-BREAK SPACE--> if  <!--NO-BREAK SPACE--></mtext></mtd> <mtd><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></mtd> </mtr> <mtr> <mtd><mi>x</mi></mtd> <mtd columnalign="left"><mtext> <!--NO-BREAK SPACE--> if  <!--NO-BREAK SPACE--></mtext></mtd> <mtd><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></mtd> </mtr> </mtable> </mrow> |
Mapping to Strict Markup
In Strict Content MathML, the container elements
piecewise
, piece
and otherwise
are mapped
to applications of the constructor symbols of the same names in the
piece1 CD. Apart from the fact that these three
elements (respectively symbols) are used together, the mapping to
Strict markup is straightforward:
Content MathML
<piecewise> <piece> <cn>0</cn> <apply><lt/><ci>x</ci><cn>0</cn></apply> </piece> <piece> <cn>1</cn> <apply><gt/><ci>x</ci><cn>1</cn></apply> </piece> <otherwise> <ci>x</ci> </otherwise> </piecewise> |
Strict Content MathML equivalent
<apply><csymbol cd="piece1">piecewise</csymbol> <apply><csymbol cd="piece1">piece</csymbol> <cn>0</cn> <apply><csymbol cd="relation1">lt</csymbol><ci>x</ci><cn>0</cn></apply> </apply> <apply><csymbol cd="piece1">piece</csymbol> <cn>1</cn> <apply><csymbol cd="relation1">gt</csymbol><ci>x</ci><cn>1</cn></apply> </apply> <apply><csymbol cd="piece1">otherwise</csymbol> <ci>x</ci> </apply> </apply> |
<quotient/>
Class | binary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | quotient |
The quotient
element represents the integer division
operator. When the operator is applied to integer arguments
a and b, the result is the "quotient of
a divided by b". That is, the quotient
of integers a and b, is the integer
q such that a = b * q +
r, with |r| less than |b| and
a * r positive. In common usage, q
is called the quotient and r is the remainder.
Content MathML
<apply><quotient/><ci>a</ci><ci>b</ci></apply> |
Sample Presentation
<mrow><mo>⌊<!--LEFT FLOOR--></mo><mi>a</mi><mo>/</mo><mi>b</mi><mo>⌋<!--RIGHT FLOOR--></mo></mrow> |
<factorial/>
Class | unary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | factorial |
This element represents the unary factorial operator on non-negative integers.
The factorial of an integer n is given by n! = n*(n-1)* ... * 1
Content MathML
<apply><factorial/><ci>n</ci></apply> |
Sample Presentation
<mrow><mi>n</mi><mo>!</mo></mrow> |
<divide/>
Class | binary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | divide |
The divide
element represents the division operator in a
number field.
Content MathML
<apply><divide/> <ci>a</ci> <ci>b</ci> </apply> |
Sample Presentation
<mrow><mi>a</mi><mo>/</mo><mi>b</mi></mrow> |
<max/>
Class | nary-minmax |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ, DomainQ |
OM Symbols | max |
The max
element denotes the maximum function, which
returns the largest of the arguments to which it is applied. Its
arguments may be explicitly specified in the enclosing
apply
element, or specified using qualifier elements
as described in Section 4.3.4.4 N-ary/Unary Operators (classes nary-minmax, nary-stats). Note that when applied to
infinite sets of arguments, no maximal argument may exist.
Content MathML
<apply><max/><cn>2</cn><cn>3</cn><cn>5</cn></apply> |
Sample Presentation
<mrow> <mi>max</mi> <mrow> <mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>}</mo> </mrow> </mrow> |
Content MathML
<apply><max/> <bvar><ci>y</ci></bvar> <condition> <apply><in/> <ci>y</ci> <interval><cn>0</cn><cn>1</cn></interval> </apply> </condition> <apply><power/><ci>y</ci><cn>3</cn></apply> </apply> |
Sample Presentation
<mrow> <mi>max</mi> <mrow> <mo>{</mo><mi>y</mi><mo>|</mo> <mrow> <msup><mi>y</mi><mn>3</mn></msup> <mo>∈<!--ELEMENT OF--></mo> <mfenced open="[" close="]"><mn>0</mn><mn>1</mn></mfenced> </mrow> <mo>}</mo> </mrow> </mrow> |
<min/>
Class | nary-minmax |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | min |
The min
element denotes the minimum function, which returns the smallest of
the arguments to which it is applied. Its arguments may be explicitly specified in the
enclosing apply
element, or specified using qualifier
elements as described in Section 4.3.4.4 N-ary/Unary Operators (classes nary-minmax, nary-stats). Note that when applied to infinite sets of arguments, no
minimal argument may exist.
Content MathML
<apply><min/><ci>a</ci><ci>b</ci></apply> |
Sample Presentation
<mrow> <mi>min</mi> <mrow><mo>{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>}</mo></mrow> </mrow> |
Content MathML
<apply><min/> <bvar><ci>x</ci></bvar> <condition> <apply><notin/><ci>x</ci><ci type="set">B</ci></apply> </condition> <apply><power/><ci>x</ci><cn>2</cn></apply> </apply> |
Sample Presentation
<mrow> <mi>min</mi> <mrow><mo>{</mo><msup><mi>x</mi><mn>2</mn></msup><mo>|</mo> <mrow><mi>x</mi><mo>∉<!--NOT AN ELEMENT OF--></mo><mi>B</mi></mrow> <mo>}</mo> </mrow> </mrow> |
<minus/>
Class | unary-arith, binary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | unary_minus, minus |
The minus
element can be used as a unary arithmetic operator
(e.g. to represent - x), or as a binary arithmetic operator
(e.g. to represent x- y).
If it is used with one argument, minus
corresponds to the unary_minus symbol.
Content MathML
<apply><minus/><cn>3</cn></apply> |
Sample Presentation
<mrow><mo>−<!--MINUS SIGN--></mo><mn>3</mn></mrow> |
If it is used with two arguments, minus
corresponds to the
minus symbol
Content MathML
<apply><minus/><ci>x</ci><ci>y</ci></apply> |
Sample Presentation
<mrow><mi>x</mi><mo>−<!--MINUS SIGN--></mo><mi>y</mi></mrow> |
In both cases, the translation to Strict Content markup is direct, as described in Rewrite: element. It is merely a matter of choosing the symbol that reflects the actual usage.
<plus/>
Class | nary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | plus |
The plus
element represents the addition operator. Its
arguments are normally specified explicitly in the enclosing
apply
element. As an n-ary commutative operator, it can
be used with qualifiers to specify arguments, however,
this is discouraged, and the sum
operator should be
used to represent such expressions instead.
Content MathML
<apply><plus/><ci>x</ci><ci>y</ci><ci>z</ci></apply> |
Sample Presentation
<mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow> |
<power/>
Class | binary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | power |
The power
element represents the exponentiation
operator. The first argument is raised to the power of the second
argument.
Content MathML
<apply><power/><ci>x</ci><cn>3</cn></apply> |
Sample Presentation
<msup><mi>x</mi><mn>3</mn></msup> |
<rem/>
Class | binary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | remainder |
The rem
element represents the modulus operator, which
returns the remainder that results from dividing the first argument by
the second. That is, when applied to integer arguments a
and b, it returns the unique integer r such that
a = b * q + r, with
|r| less than |b| and a *
r positive.
Content MathML
<apply><rem/><ci> a </ci><ci> b </ci></apply> |
Sample Presentation
<mrow><mi>a</mi><mo>mod</mo><mi>b</mi></mrow> |
<times/>
Class | nary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | times |
The times
element represents the n-ary multiplication operator. Its
arguments are normally specified explicitly in the enclosing
apply
element. As an n-ary commutative operator, it can
be used with qualifiers to specify arguments by rule, however,
this is discouraged, and the product
operator should be
used to represent such expressions instead.
Content MathML
<apply><times/><ci>a</ci><ci>b</ci></apply> |
Sample Presentation
<mrow><mi>a</mi><mo>⁢<!--INVISIBLE TIMES--></mo><mi>b</mi></mrow> |
<root/>
Class | unary-arith, binary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | degree |
OM Symbols | root |
The root
element is used to extract roots. The kind of root to be taken is
specified by a "degree" element, which should be given as the second child of
the apply
element enclosing the root
element. Thus, square roots
correspond to the case where degree
contains the value 2, cube roots
correspond to 3, and so on. If no degree
is present, a default value of 2 is
used.
Content MathML
<apply><root/> <degree><ci type="integer">n</ci></degree> <ci>a</ci> </apply> |
Sample Presentation
<mroot><mi>a</mi><mi>n</mi></mroot> |
Mapping to Strict Content Markup
In Strict Content markup, the root symbol is always used with two arguments, with the second indicating the degree of the root being extracted.
Content MathML
<apply><root/><ci>x</ci></apply> |
Strict Content MathML equivalent
<apply><csymbol cd="arith1">root</csymbol> <ci>x</ci> <cn type="integer">2</cn> </apply> |
Content MathML
<apply><root/> <degree><ci type="integer">n</ci></degree> <ci>a</ci> </apply> |
Strict Content MathML equivalent
<apply><csymbol cd="arith1">root</csymbol> <ci>a</ci> <cn type="integer">n</cn> </apply> |
<gcd/>
Class | nary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | gcd |
The gcd
element represents the n-ary operator which returns
the greatest common divisor of its arguments. Its
arguments may be explicitly specified in the enclosing
apply
element, or specified by rule as described in
Section 4.3.4.1 N-ary Operators (classes nary-arith, nary-functional, nary-logical,
nary-linalg, nary-set, nary-constructor).
Content MathML
<apply><gcd/><ci>a</ci><ci>b</ci><ci>c</ci></apply> |
Sample Presentation
<mrow> <mi>gcd</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>a</mi><mi>b</mi><mi>c</mi></mfenced> </mrow> |
This default rendering is English-language locale specific: other locales may have different default renderings.
When the gcd
element is applied to an explicit list of arguments, the
translation to Strict Content markup is direct, using the gcd symbol, as described in Rewrite: element. However, when
qualifiers are used, the equivalent Strict markup is computed via
Rewrite: n-ary domainofapplication.
<and/>
Class | nary-logical |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | and |
The and
element represents the logical "and" function which is
an n-ary function taking Boolean arguments and returning a Boolean value. It is true if
all arguments are true, and false otherwise. Its arguments may be explicitly specified
in the enclosing apply
element, or specified by rule as described in Section 4.3.4.1 N-ary Operators (classes nary-arith, nary-functional, nary-logical,
nary-linalg, nary-set, nary-constructor).
Content MathML
<apply><and/><ci>a</ci><ci>b</ci></apply> |
Sample Presentation
<mrow><mi>a</mi><mo>∧<!--LOGICAL AND--></mo><mi>b</mi></mrow> |
Content MathML
<apply><and/> <bvar><ci>i</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><ci>n</ci></uplimit> <apply><gt/><apply><selector/><ci>a</ci><ci>i</ci></apply><cn>0</cn></apply> </apply> |
Strict Content MathML
<apply><csymbol cd="fns2">apply_to_list</csymbol> <csymbol cd="logic1">and</csymbol> <apply><csymbol cd="list1">map</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>i</ci></bvar> <apply><csymbol cd="relation1">gt</csymbol> <apply><csymbol cd="linalg1">vector_selector</csymbol> <ci>i</ci> <ci>a</ci> </apply> <cn>0</cn> </apply> </bind> <apply><csymbol cd="interval1">integer_interval</csymbol> <cn type="integer">0</cn> <ci>n</ci> </apply> </apply> </apply> |
Sample Presentation
<mrow> <munderover> <mo>⋀<!--N-ARY LOGICAL AND--></mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi> </munderover> <mrow> <mo>(</mo> <msub><mi>a</mi><mi>i</mi></msub> <mo>></mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> |
<or/>
Class | nary-logical |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | or |
The or
element represents the logical "or" function. It is
true if any of the arguments are true, and false otherwise.
Content MathML
<apply><or/><ci>a</ci><ci>b</ci></apply> |
Sample Presentation
<mrow><mi>a</mi><mo>∨<!--LOGICAL OR--></mo><mi>b</mi></mrow> |
<xor/>
Class | nary-logical |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | xor |
The xor
element represents the logical "xor"
function. It is true if there are an odd number of true arguments or
false otherwise.
Content MathML
<apply><xor/><ci>a</ci><ci>b</ci></apply> |
Sample Presentation
<mrow><mi>a</mi><mo>xor</mo><mi>b</mi></mrow> |
<not/>
Class | unary-logical |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | not |
The not
element represents the logical not function
which takes one Boolean argument, and returns the opposite Boolean
value.
Content MathML
<apply><not/><ci>a</ci></apply> |
Sample Presentation
<mrow><mo>¬<!--NOT SIGN--></mo><mi>a</mi></mrow> |
<implies/>
Class | binary-logical |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | implies |
The implies
element represents the logical implication
function which takes two Boolean expressions as arguments. It
evaluates to false if the first argument is true and the second
argument is false, otherwise it evaluates to true.
Content MathML
<apply><implies/><ci>A</ci><ci>B</ci></apply> |
Sample Presentation
<mrow><mi>A</mi><mo>⇒<!--RIGHTWARDS DOUBLE ARROW--></mo><mi>B</mi></mrow> |
<forall/>
Class | quantifier |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | forall, implies |
The forall
element represents the universal ("for all")
quantifier which takes one or more bound variables, and an
argument which specifies the assertion being quantified.
In addition, condition
or other qualifiers may be used as
described in Section 4.3.4.8 Quantifiers (class quantifier) to limit the domain
of the bound variables.
Content MathML
<bind><forall/> <bvar><ci>x</ci></bvar> <apply><eq/> <apply><minus/><ci>x</ci><ci>x</ci></apply> <cn>0</cn> </apply> </bind> |
Sample Presentation
<mrow> <mo>∀<!--FOR ALL--></mo> <mi>x</mi> <mo>.</mo> <mfenced> <mrow> <mrow><mi>x</mi><mo>−<!--MINUS SIGN--></mo><mi>x</mi></mrow> <mo>=</mo> <mn>0</mn> </mrow> </mfenced> </mrow> |
Mapping to Strict Markup
When the forall
element is used with a condition
qualifier the
strict equivalent is constructed with the help of logical implication by the rule
Rewrite: quantifier. Thus
<bind><forall/> <bvar><ci>p</ci></bvar> <bvar><ci>q</ci></bvar> <condition> <apply><and/> <apply><in/><ci>p</ci><rationals/></apply> <apply><in/><ci>q</ci><rationals/></apply> <apply><lt/><ci>p</ci><ci>q</ci></apply> </apply> </condition> <apply><lt/> <ci>p</ci> <apply><power/><ci>q</ci><cn>2</cn></apply> </apply> </bind> |
translates to
<bind><csymbol cd="quant1">forall</csymbol> <bvar><ci>p</ci></bvar> <bvar><ci>q</ci></bvar> <apply><csymbol cd="logic1">implies</csymbol> <apply><csymbol cd="logic1">and</csymbol> <apply><csymbol cd="set1">in</csymbol> <ci>p</ci> <csymbol cd="setname1">Q</csymbol> </apply> <apply><csymbol cd="set1">in</csymbol> <ci>q</ci> <csymbol cd="setname1">Q</csymbol> </apply> <apply><csymbol cd="relation1">lt</csymbol><ci>p</ci><ci>q</ci></apply> </apply> <apply><csymbol cd="relation1">lt</csymbol> <ci>p</ci> <apply><csymbol cd="arith1">power</csymbol> <ci>q</ci> <cn>2</cn> </apply> </apply> </apply> </bind> |
Sample Presentation
<mrow> <mo>∀<!--FOR ALL--></mo> <mrow> <mrow><mi>p</mi><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">Q</mi></mrow> <mo>∧<!--LOGICAL AND--></mo> <mrow><mi>q</mi><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">Q</mi></mrow> <mo>∧<!--LOGICAL AND--></mo> <mrow><mo>(</mo><mi>p</mi><mo><</mo><mi>q</mi><mo>)</mo></mrow> </mrow> <mo>.</mo> <mfenced> <mrow><mi>p</mi><mo><</mo><msup><mi>q</mi><mn>2</mn></msup></mrow> </mfenced> </mrow> |
<mrow> <mo>∀<!--FOR ALL--></mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mo>.</mo> <mfenced> <mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">Q</mi> </mrow> <mo>∧<!--LOGICAL AND--></mo> <mrow> <mi>q</mi><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">Q</mi> </mrow> <mo>∧<!--LOGICAL AND--></mo> <mrow><mo>(</mo><mi>p</mi><mo><</mo><mi>q</mi><mo>)</mo></mrow> <mo>)</mo> </mrow> <mo>⇒<!--RIGHTWARDS DOUBLE ARROW--></mo> <mrow> <mo>(</mo> <mi>p</mi> <mo><</mo> <msup><mi>q</mi><mn>2</mn></msup> <mo>)</mo> </mrow> </mrow> </mfenced> </mrow> |
<exists/>
Class | quantifier |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | exists, and |
The exists
element represents the existential ("there exists")
quantifier which takes one or more bound variables, and an
argument which specifies the assertion being quantified. In
addition, condition
or other qualifiers may be used as
described in Section 4.3.4.8 Quantifiers (class quantifier) to limit the domain
of the bound variables.
Content MathML
<bind><exists/> <bvar><ci>x</ci></bvar> <apply><eq/> <apply><ci>f</ci><ci>x</ci></apply> <cn>0</cn> </apply> </bind> |
Sample Presentation
<mrow> <mo>∃<!--THERE EXISTS--></mo> <mi>x</mi> <mo>.</mo> <mfenced> <mrow> <mrow><mi>f</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi></mfenced></mrow> <mo>=</mo> <mn>0</mn> </mrow> </mfenced> </mrow> |
Content MathML
<apply><exists/> <bvar><ci>x</ci></bvar> <domainofapplication> <integers/> </domainofapplication> <apply><eq/> <apply><ci>f</ci><ci>x</ci></apply> <cn>0</cn> </apply> </apply> |
Strict MathML equivalent:
<bind><csymbol cd="quant1">exists</csymbol> <bvar><ci>x</ci></bvar> <apply><csymbol cd="logic1">and</csymbol> <apply><csymbol cd="set1">in</csymbol> <ci>x</ci> <csymbol cd="setname1">Z</csymbol> </apply> <apply><csymbol cd="relation1">eq</csymbol> <apply><ci>f</ci><ci>x</ci></apply> <cn>0</cn> </apply> </apply> </bind> |
Sample Presentation
<mrow> <mo>∃<!--THERE EXISTS--></mo> <mi>x</mi> <mo>.</mo> <mfenced separators=""> <mrow><mi>x</mi><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">Z</mi></mrow> <mo>∧<!--LOGICAL AND--></mo> <mrow> <mrow><mi>f</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi></mfenced></mrow> <mo>=</mo> <mn>0</mn> </mrow> </mfenced> </mrow> |
<abs/>
Class | unary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | abs |
The abs
element represents the absolute value
function. The argument should be numerically valued. When the
argument is a complex number, the absolute value is often referred
to as the modulus.
Content MathML
<apply><abs/><ci>x</ci></apply> |
Sample Presentation
<mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow> |
<conjugate/>
Class | unary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | conjugate |
The conjugate
element represents the function defined
over the complex numbers with returns the complex conjugate of
its argument.
Content MathML
<apply><conjugate/> <apply><plus/> <ci>x</ci> <apply><times/><cn>ⅈ<!--DOUBLE-STRUCK ITALIC SMALL I--></cn><ci>y</ci></apply> </apply> </apply> |
Sample Presentation
<mover> <mrow> <mi>x</mi> <mo>+</mo> <mrow><mn>ⅈ<!--DOUBLE-STRUCK ITALIC SMALL I--></mn><mo>⁢<!--INVISIBLE TIMES--></mo><mi>y</mi></mrow> </mrow> <mo>¯<!--MACRON--></mo> </mover> |
<arg/>
Class | unary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | argument |
The arg
element represents the unary function which
returns the angular argument of a complex number, namely the
angle which a straight line drawn from the number to zero makes
with the real line (measured anti-clockwise).
Content MathML
<apply><arg/> <apply><plus/> <ci> x </ci> <apply><times/><imaginaryi/><ci>y</ci></apply> </apply> </apply> |
Sample Presentation
<mrow> <mi>arg</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced> <mrow> <mi>x</mi> <mo>+</mo> <mrow><mi>i</mi><mo>⁢<!--INVISIBLE TIMES--></mo><mi>y</mi></mrow> </mrow> </mfenced> </mrow> |
<real/>
Class | unary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | real |
The real
element represents the unary operator used to
construct an expression representing the "real" part of a
complex number, that is, the x component in x + iy.
Content MathML
<apply><real/> <apply><plus/> <ci>x</ci> <apply><times/><imaginaryi/><ci>y</ci></apply> </apply> </apply> |
Sample Presentation
<mrow> <mo>ℛ<!--SCRIPT CAPITAL R--></mo> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced> <mrow> <mi>x</mi> <mo>+</mo> <mrow><mi>i</mi><mo>⁢<!--INVISIBLE TIMES--></mo><mi>y</mi></mrow> </mrow> </mfenced> </mrow> |
<imaginary/>
Class | unary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | imaginary |
The imaginary
element represents the unary operator used to
construct an expression representing the "imaginary" part of a
complex number, that is, the y component in x + iy.
Content MathML
<apply><imaginary/> <apply><plus/> <ci>x</ci> <apply><times/><imaginaryi/><ci>y</ci></apply> </apply> </apply> |
Sample Presentation
<mrow> <mo>ℑ<!--BLACK-LETTER CAPITAL I--></mo> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced> <mrow> <mi>x</mi> <mo>+</mo> <mrow><mi>i</mi><mo>⁢<!--INVISIBLE TIMES--></mo><mi>y</mi></mrow> </mrow> </mfenced> </mrow> |
<lcm/>
Class | nary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | lcm |
The lcm
element represents the n-ary operator used to
construct an expression which represents the least common multiple
of its arguments. If no argument is provided, the lcm is 1. If one
argument is provided, the lcm is that argument. The least common
multiple of x and 1 is x.
Content MathML
<apply><lcm/><ci>a</ci><ci>b</ci><ci>c</ci></apply> |
Sample Presentation
<mrow> <mi>lcm</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>a</mi><mi>b</mi><mi>c</mi></mfenced> </mrow> |
This default rendering is English-language locale specific: other locales may have different default renderings.
<floor/>
Class | unary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | floor |
The floor
element represents the operation that rounds
down (towards negative infinity) to the nearest integer. This
function takes one real number as an argument and returns an
integer.
Content MathML
<apply><floor/><ci>a</ci></apply> |
Sample Presentation
<mrow><mo>⌊<!--LEFT FLOOR--></mo><mi>a</mi><mo>⌋<!--RIGHT FLOOR--></mo></mrow> |
<ceiling/>
Class | unary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | ceiling |
The ceiling
element represents the operation that rounds
up (towards positive infinity) to the nearest integer. This function
takes one real number as an argument and returns an integer.
Content MathML
<apply><ceiling/><ci>a</ci></apply> |
Sample Presentation
<mrow><mo>⌈<!--LEFT CEILING--></mo><mi>a</mi><mo>⌉<!--RIGHT CEILING--></mo></mrow> |
<eq/>
Class | nary-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | eq |
The eq
elements represents the equality relation. While equality is a binary relation,
eq
may be used with more than two arguments, denoting a chain
of equalities, as described in Section 4.3.4.3 N-ary Relations (classes nary-reln, nary-set-reln).
Content MathML
<apply><eq/> <cn type="rational">2<sep/>4</cn> <cn type="rational">1<sep/>2</cn> </apply> |
Sample Presentation
<mrow> <mrow><mn>2</mn><mo>/</mo><mn>4</mn></mrow> <mo>=</mo> <mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow> </mrow> |
<neq/>
Class | binary-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | neq |
The neq
element represents the binary inequality
relation, i.e. the relation "not equal to" which returns true unless
the two arguments are equal.
Content MathML
<apply><neq/><cn>3</cn><cn>4</cn></apply> |
Sample Presentation
<mrow><mn>3</mn><mo>≠<!--NOT EQUAL TO--></mo><mn>4</mn></mrow> |
<gt/>
Class | nary-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | gt |
The gt
element represents the "greater than" function
which returns true if the first argument is greater than the second, and
returns false otherwise. While this is a binary relation,
gt
may be used with more than two arguments, denoting a chain
of inequalities, as described in Section 4.3.4.3 N-ary Relations (classes nary-reln, nary-set-reln).
Content MathML
<apply><gt/><cn>3</cn><cn>2</cn></apply> |
Sample Presentation
<mrow><mn>3</mn><mo>></mo><mn>2</mn></mrow> |
<lt/>
Class | nary-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | lt |
The lt
element represents the "less than" function
which returns true if the first argument is less than the second, and
returns false otherwise. While this is a binary relation,
lt
may be used with more than two arguments, denoting a chain
of inequalities, as described in Section 4.3.4.3 N-ary Relations (classes nary-reln, nary-set-reln).
Content MathML
<apply><lt/><cn>2</cn><cn>3</cn><cn>4</cn></apply> |
Sample Presentation
<mrow><mn>2</mn><mo><</mo><mn>3</mn><mo><</mo><mn>4</mn></mrow> |
<geq/>
Class | nary-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | geq |
The geq
element represents the "greater than or equal to" function
which returns true if the first argument is greater than or equal to
the second, and returns false otherwise. While this is a binary relation,
geq
may be used with more than two arguments, denoting a chain
of inequalities, as described in Section 4.3.4.3 N-ary Relations (classes nary-reln, nary-set-reln).
Content MathML
<apply><geq/><cn>4</cn><cn>3</cn><cn>3</cn></apply> |
Strict Content MathML
<apply><csymbol cd="fns2">predicate_on_list</csymbol> <csymbol cd="reln1">geq</csymbol> <apply><csymbol cd="list1">list</csymbol> <cn>4</cn><cn>3</cn><cn>3</cn> </apply> </apply> |
Sample Presentation
<mrow><mn>4</mn><mo>≥<!--GREATER-THAN OR EQUAL TO--></mo><mn>3</mn><mo>≥<!--GREATER-THAN OR EQUAL TO--></mo><mn>3</mn></mrow> |
<leq/>
Class | nary-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | leq |
The leq
element represents the "less than or equal to" function
which returns true if the first argument is less than or equal to
the second, and returns false otherwise. While this is a binary relation,
leq
may be used with more than two arguments, denoting a chain
of inequalities, as described in Section 4.3.4.3 N-ary Relations (classes nary-reln, nary-set-reln).
Content MathML
<apply><leq/><cn>3</cn><cn>3</cn><cn>4</cn></apply> |
Sample Presentation
<mrow><mn>3</mn><mo>≤<!--LESS-THAN OR EQUAL TO--></mo><mn>3</mn><mo>≤<!--LESS-THAN OR EQUAL TO--></mo><mn>4</mn></mrow> |
<equivalent/>
Class | binary-logical |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | equivalent |
The equivalent
element represents the relation that
asserts two Boolean expressions are logically equivalent,
that is have the same Boolean value for any inputs.
Content MathML
<apply><equivalent/> <ci>a</ci> <apply><not/><apply><not/><ci>a</ci></apply></apply> </apply> |
Sample Presentation
<mrow> <mi>a</mi> <mo>≡<!--IDENTICAL TO--></mo> <mrow><mo>¬<!--NOT SIGN--></mo><mrow><mo>¬<!--NOT SIGN--></mo><mi>a</mi></mrow></mrow> </mrow> |
<approx/>
Class | binary-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | approx |
The approx
element represent the relation that asserts
the approximate equality of its arguments.
Content MathML
<apply><approx/> <pi/> <cn type="rational">22<sep/>7</cn> </apply> |
Sample Presentation
<mrow> <mi>π<!--GREEK SMALL LETTER PI--></mi> <mo>≃<!--ASYMPTOTICALLY EQUAL TO--></mo> <mrow><mn>22</mn><mo>/</mo><mn>7</mn></mrow> </mrow> |
<factorof/>
Class | binary-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | factorof |
The factorof
element is used to indicate the
mathematical relationship that the first argument "is a factor of"
the second. This relationship is true if and only
if b mod a = 0.
Content MathML
<apply><factorof/><ci>a</ci><ci>b</ci></apply> |
Sample Presentation
<mrow><mi>a</mi><mo>|</mo><mi>b</mi></mrow> |
<int/>
Class | int |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | int defint |
The int
element is the operator element for a definite or indefinite integral
over a function or a definite over an expression with a bound variable.
Content MathML
<apply><eq/> <apply><int/><sin/></apply> <cos/> </apply> |
Sample Presentation
<mrow><mrow><mi>∫<!--INTEGRAL--></mi><mi>sin</mi></mrow><mo>=</mo><mi>cos</mi></mrow> |
Content MathML
<apply><int/> <interval><ci>a</ci><ci>b</ci></interval> <cos/> </apply> |
Sample Presentation
<mrow> <msubsup><mi>∫<!--INTEGRAL--></mi><mi>a</mi><mi>b</mi></msubsup><mi>cos</mi> </mrow> |
The int
element can also be used with bound variables serving as the
integration variables.
Content MathML
Here, definite integrals are indicated by providing qualifier elements specifying a
domain of integration (here a lowlimit
/uplimit
pair). This is perhaps
the most "standard" representation of this integral:
<apply><int/> <bvar><ci>x</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><cn>1</cn></uplimit> <apply><power/><ci>x</ci><cn>2</cn></apply> </apply> |
Sample Presentation
<mrow> <msubsup><mi>∫<!--INTEGRAL--></mi><mn>0</mn><mn>1</mn></msubsup> <msup><mi>x</mi><mn>2</mn></msup> <mi>d</mi> <mi>x</mi> </mrow> |
Mapping to Strict Markup
As an indefinite integral applied to a function, the int
element corresponds to
the int symbol from the calculus1 content
dictionary. As a definite integral applied to a function, the int
element
corresponds to the defint symbol
from the calculus1 content dictionary.
When no bound variables are present, the translation of an indefinite integral to Strict Content Markup is straight forward. When bound variables are present, the following rule should be used.
Translate an indefinite integral, where
is an
arbitrary expression involving the bound variable(s)
<apply><int/> <bvar></bvar> </apply> |
to the expression
<apply> <apply><csymbol cd="calculus1">int</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar></bvar> </bind> </apply> </apply> |
Note that as x is not bound in the original indefinite integral, the integrated function is applied to the variable x making it an explicit free variable in Strict Content Markup expression, even though it is bound in the subterm used as an argument to int.
For instance, the expression
<apply><int/> <bvar><ci>x</ci></bvar> <apply><cos/><ci>x</ci></apply> </apply> |
has the Strict Content MathML equivalent
<apply> <apply><csymbol cd="calculus1">int</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>x</ci></bvar> <apply><cos/><ci>x</ci></apply> </bind> </apply> <ci>x</ci> </apply> |
For a definite integral without bound variables, the translation is also straightforward.
For instance, the integral of a differential form f over an arbitrary domain C represented as
<apply><int/> <domainofapplication><ci>C</ci></domainofapplication> <ci>f</ci> </apply> |
is equivalent to the Strict Content MathML:
<apply><csymbol cd="calculus1">defint</csymbol><ci>C</ci><ci>f</ci></apply> |
Note, however, the additional remarks on the translations of other kinds of qualifiers that may be used to specify a domain of integration in the rules for definite integrals following.
When bound variables are present, the situation is more complicated in general, and the following rules are used.
Translate a definite integral, where
is an
arbitrary expression involving the bound variable(s)
<apply><int/> <bvar></bvar> <domainofapplication> </domainofapplication> </apply> |
to the expression
<apply><csymbol cd="calculus1">defint</csymbol><bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> </bind> </apply> |
But the definite integral with an lowlimit
/uplimit
pair carries the
strong intuition that the range of integration is oriented, and thus swapping lower and
upper limits will change the sign of the result. To accommodate this, use the following special
translation rule:
<apply><int/> <bvar></bvar> <lowlimit> </lowlimit> <uplimit> </uplimit> </apply> |
where
is an expression in the variable x
is translated to to the expression:
<apply><csymbol cd="calculus1">defint</csymbol> <apply><csymbol cd="interval1">oriented_interval</csymbol></apply> <bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> </bind> </apply> |
The oriented_interval
symbol is also used when translating the interval
qualifier, when it is used to specify the domain of integration.
Integration is assumed to proceed from the left endpoint to the
right endpoint.
The case for multiple integrands is treated analogously.
Note that use of the condition
qualifier also
requires special treatment. In particular, it extends to multivariate domains by
using extra bound variables and a domain corresponding to a cartesian product as in:
<bind><int/> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <condition> <apply><and/> <apply><leq/><cn>0</cn><ci>x</ci></apply> <apply><leq/><ci>x</ci><cn>1</cn></apply> <apply><leq/><cn>0</cn><ci>y</ci></apply> <apply><leq/><ci>y</ci><cn>1</cn></apply> </apply> </condition> <apply><times/> <apply><power/><ci>x</ci><cn>2</cn></apply> <apply><power/><ci>y</ci><cn>3</cn></apply> </apply> </bind> |
Strict Content MathML equivalent
<apply><csymbol cd="calculus1">defint</csymbol> <apply><csymbol cd="set1">suchthat</csymbol> <apply><csymbol cd="set1">cartesianproduct</csymbol> <csymbol cd="setname1">R</csymbol> <csymbol cd="setname1">R</csymbol> </apply> <apply><csymbol cd="logic1">and</csymbol> <apply><csymbol cd="arith1">leq</csymbol><cn>0</cn><ci>x</ci></apply> <apply><csymbol cd="arith1">leq</csymbol><ci>x</ci><cn>1</cn></apply> <apply><csymbol cd="arith1">leq</csymbol><cn>0</cn><ci>y</ci></apply> <apply><csymbol cd="arith1">leq</csymbol><ci>y</ci><cn>1</cn></apply> </apply> <bind><csymbol cd="fns11">lambda</csymbol> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <apply><csymbol cd="arith1">times</csymbol> <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><cn>2</cn></apply> <apply><csymbol cd="arith1">power</csymbol><ci>y</ci><cn>3</cn></apply> </apply> </bind> </apply> </apply> |
<diff/>
Class | Differential-Operator |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | diff |
The diff
element is the differentiation operator element for functions or
expressions of a single variable. It may be applied directly to an actual function
thereby denoting a function which is the derivative of the original function, or it can be
applied to an expression involving a single variable.
Content MathML
<apply><diff/><ci>f</ci></apply> |
Sample Presentation
<msup><mi>f</mi><mo>′<!--PRIME--></mo></msup> |
Content MathML
<apply><eq/> <apply><diff/> <bvar><ci>x</ci></bvar> <apply><sin/><ci>x</ci></apply> </apply> <apply><cos/><ci>x</ci></apply> </apply> |
Sample Presentation
<mrow> <mfrac> <mrow><mi>d</mi><mrow><mi>sin</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow></mrow> <mrow><mi>d</mi><mi>x</mi></mrow> </mfrac> <mo>=</mo> <mrow><mi>cos</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow> </mrow> |
The bvar
element may also contain a degree
element, which specifies
the order of the derivative to be taken.
Content MathML
<apply><diff/> <bvar><ci>x</ci><degree><cn>2</cn></degree></bvar> <apply><power/><ci>x</ci><cn>4</cn></apply> </apply> |
Sample Presentation
<mfrac> <mrow> <msup><mi>d</mi><mn>2</mn></msup> <msup><mi>x</mi><mn>4</mn></msup> </mrow> <mrow><mi>d</mi><msup><mi>x</mi><mn>2</mn></msup></mrow> </mfrac> |
Mapping to Strict Markup
For the translation to strict Markup it is crucial to realize that in the expression case, the variable is actually not bound by the differentiation operator.
Translate an expression
<apply><diff/> <bvar></bvar> </apply> |
where
is an
expression in the variable x
to the expression
<apply> <apply><csymbol cd="calculus1">diff</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar></bvar> </bind> </apply> </apply> |
Note that the differentiated function is applied to the variable x making its status as a free variable explicit in strict markup. Thus the strict equivalent of
<apply><diff/> <bvar><ci>x</ci></bvar> <apply><sin/><ci>x</ci></apply> </apply> |
is
<apply> <apply><csymbol cd="calculus1">diff</csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>x</ci></bvar> <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply> </bind> </apply> <ci>x</ci> </apply> |
If the bvar
element contains a degree
element, use the
nthdiff
symbol.
<apply><diff/> <bvar><degree> </degree></bvar> </apply> |
where
is an
is an expression in the variable x
is translated to to the expression:
<apply> <apply><csymbol cd="calculus1">nthdiff</csymbol><bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> </bind> </apply> </apply> |
For example
<apply><diff/> <bvar><degree><cn>2</cn></degree><ci>x</ci></bvar> <apply><sin/><ci>x</ci></apply> </apply> |
Strict Content MathML equivalent
<apply> <apply><csymbol cd="calculus1">nthdiff</csymbol> <cn>2</cn> <bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>x</ci></bvar> <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply> </bind> </apply> <ci>x</ci> </apply> |
<partialdiff/>
Class | partialdiff |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | partialdiff partialdiffdegree |
The partialdiff
element is the partial differentiation operator element for
functions or expressions in several variables.
For the case of partial differentiation of a function, the
containing partialdiff
takes two arguments: firstly a list of
indices indicating by position which function arguments are involved in
constructing the partial derivatives, and secondly the actual function
to be partially differentiated. The indices may be repeated.
Content MathML
<apply><partialdiff/> <list><cn>1</cn><cn>1</cn><cn>3</cn></list> <ci type="function">f</ci> </apply> |
Sample Presentation
<mrow> <msub> <mi>D</mi> <mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn></mrow> </msub> <mi>f</mi> </mrow> |
Content MathML
<apply><partialdiff/> <list><cn>1</cn><cn>1</cn><cn>3</cn></list> <lambda> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <bvar><ci>z</ci></bvar> <apply><ci>f</ci><ci>x</ci><ci>y</ci><ci>z</ci></apply> </lambda> </apply> |
Sample Presentation
<mfrac> <mrow> <msup><mo>∂<!--PARTIAL DIFFERENTIAL--></mo><mn>3</mn></msup> <mrow> <mi>f</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi><mi>y</mi><mi>z</mi></mfenced> </mrow> </mrow> <mrow> <mrow><mo>∂<!--PARTIAL DIFFERENTIAL--></mo><msup><mi>x</mi><mn>2</mn></msup></mrow> <mrow><mo>∂<!--PARTIAL DIFFERENTIAL--></mo><mi>z</mi></mrow> </mrow> </mfrac> |
In the case of algebraic expressions, the bound variables are given by bvar
elements, which are children of the containing apply
element. The bvar
elements may also contain degree
element, which specify the order of the partial
derivative to be taken in that variable.
Content MathML
<apply><partialdiff/> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <apply><ci type="function">f</ci><ci>x</ci><ci>y</ci></apply> </apply> |
Sample Presentation
<mfrac> <mrow> <msup><mo>∂<!--PARTIAL DIFFERENTIAL--></mo><mn>2</mn></msup> <mrow> <mi>f</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>x</mi><mi>y</mi></mfenced> </mrow> </mrow> <mrow> <mrow><mo>∂<!--PARTIAL DIFFERENTIAL--></mo><mi>x</mi></mrow> <mrow><mo>∂<!--PARTIAL DIFFERENTIAL--></mo><mi>y</mi></mrow> </mrow> </mfrac> |
Where a total degree of differentiation must be specified, this is
indicated by use of a degree
element at the top level,
i.e. without any associated bvar
, as a child of the
containing apply
element.
Content MathML
<apply><partialdiff/> <bvar><ci>x</ci><degree><ci>m</ci></degree></bvar> <bvar><ci>y</ci><degree><ci>n</ci></degree></bvar> <degree><ci>k</ci></degree> <apply><ci type="function">f</ci> <ci>x</ci> <ci>y</ci> </apply> </apply> |
Sample Presentation
<mfrac> <mrow> <msup><mo>∂<!--PARTIAL DIFFERENTIAL--></mo><mi>k</mi></msup> <mrow> <mi>f</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi><mi>y</mi></mfenced> </mrow> </mrow> <mrow> <mrow><mo>∂<!--PARTIAL DIFFERENTIAL--></mo><msup><mi>x</mi><mi>m</mi></msup></mrow> <mrow><mo>∂<!--PARTIAL DIFFERENTIAL--></mo><msup><mi>y</mi><mi>n</mi></msup></mrow> </mrow> </mfrac> |
Mapping to Strict Markup
When applied to a function, the partialdiff
element
corresponds to the partialdiff
symbol from the calculus1 content dictionary. No special
rules are necessary as the two arguments of partialdiff
translate directly to the two arguments of
partialdiff.
If partialdiff
is used with an expression and
bvar
qualifiers it is rewritten to
Strict Content MathML using the
partialdiffdegree symbol.
<apply><partialdiff/> <bvar><degree> </degree></bvar> <bvar> <degree> </degree></bvar> <degree> </degree> </apply> |
is an
arbitrary expression involving the bound variables.
<apply> <apply><csymbol cd="calculus1">partialdiffdegree</csymbol> <apply><csymbol cd="list1">list</csymbol></apply> <bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> <bvar> </bvar> </bind> </apply> </apply> |
If any of the bound variables do not use a degree
qualifier,
<cn>1</cn>
should be used in place of the degree.
If the original expression did not use the total degree qualifier then
the second argument to partialdiffdegree
should be the sum of the degrees, for example
<apply><csymbol cd="arith1">plus</csymbol></apply> |
With this rule, the expression
<apply><partialdiff/> <bvar><ci>x</ci><degree><ci>n</ci></degree></bvar> <bvar><ci>y</ci><degree><ci>m</ci></degree></bvar> <apply><sin/> <apply><times/><ci>x</ci><ci>y</ci></apply> </apply> </apply> |
is translated into
<apply> <apply><csymbol cd="calculus1">partialdiffdegree</csymbol> <apply><csymbol cd="list1">list</csymbol> <ci>n</ci><ci>m</ci> </apply> <apply><csymbol cd="arith1">plus</csymbol> <ci>n</ci><ci>m</ci> </apply> <bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <apply><csymbol cd="transc1">sin</csymbol> <apply><csymbol cd="arith1">times</csymbol> <ci>x</ci><ci>y</ci> </apply> </apply> </bind> <ci>x</ci> <ci>y</ci> </apply> </apply> |
<divergence/>
Class | unary-veccalc |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | divergence |
The divergence
element is the vector calculus divergence
operator, often called div. It represents the divergence function
which takes one argument which should be a vector of scalar-valued
functions, intended to represent a vector-valued function, and returns
the scalar-valued function giving the divergence of the argument.
Content MathML
<apply><divergence/><ci>a</ci></apply> |
Sample Presentation
<mrow><mi>div</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>a</mi></mfenced></mrow> |
Content MathML
<apply><divergence/> <ci type="vector">E</ci> </apply> |
Sample Presentation
<mrow><mi>div</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>E</mi></mfenced></mrow> |
<mrow><mo>∇<!--NABLA--></mo><mo>⋅<!--DOT OPERATOR--></mo><mi>E</mi></mrow> |
The functions defining the coordinates may be defined implicitly as expressions defined in terms of the coordinate names, in which case the coordinate names must be provided as bound variables.
Content MathML
<apply><divergence/> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <bvar><ci>z</ci></bvar> <vector> <apply><plus/><ci>x</ci><ci>y</ci></apply> <apply><plus/><ci>x</ci><ci>z</ci></apply> <apply><plus/><ci>z</ci><ci>y</ci></apply> </vector> </apply> |
Sample Presentation
<mrow> <mi>div</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mo>(</mo> <mtable> <mtr><mtd> <mi>x</mi> <mo>↦<!--RIGHTWARDS ARROW FROM BAR--></mo> <mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow> </mtd></mtr> <mtr><mtd> <mi>y</mi> <mo>↦<!--RIGHTWARDS ARROW FROM BAR--></mo> <mrow><mi>x</mi><mo>+</mo><mi>z</mi></mrow> </mtd></mtr> <mtr><mtd> <mi>z</mi> <mo>↦<!--RIGHTWARDS ARROW FROM BAR--></mo> <mrow><mi>z</mi><mo>+</mo><mi>y</mi></mrow> </mtd></mtr> </mtable> <mo>)</mo> </mrow> |
<grad/>
Class | unary-veccalc |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | grad |
The grad
element is the vector calculus gradient operator, often called
grad. It is used to represent the grad function, which takes one
argument which should be a scalar-valued function and returns a
vector of functions.
Content MathML
<apply><grad/><ci type="function">f</ci></apply> |
Sample Presentation
<mrow><mi>grad</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>f</mi></mfenced></mrow> |
<mrow><mo>∇<!--NABLA--></mo><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>f</mi></mfenced></mrow> |
The functions defining the coordinates may be defined implicitly as expressions defined in terms of the coordinate names, in which case the coordinate names must be provided as bound variables.
Content MathML
<apply><grad/> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <bvar><ci>z</ci></bvar> <apply><times/><ci>x</ci><ci>y</ci><ci>z</ci></apply> </apply> |
Sample Presentation
<mrow> <mi>grad</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mrow> <mo>(</mo> <mfenced><mi>x</mi><mi>y</mi><mi>z</mi></mfenced> <mo>↦<!--RIGHTWARDS ARROW FROM BAR--></mo> <mrow> <mi>x</mi><mo>⁢<!--INVISIBLE TIMES--></mo><mi>y</mi><mo>⁢<!--INVISIBLE TIMES--></mo><mi>z</mi> </mrow> <mo>)</mo> </mrow> </mrow> |
<curl/>
Class | unary-veccalc |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | curl |
The curl
element is used to represent the curl function
of vector calculus. It takes one argument which should be a vector
of scalar-valued functions, intended to represent a vector-valued
function, and returns a vector of functions.
Content MathML
<apply><curl/><ci>a</ci></apply> |
Sample Presentation
<mrow><mi>curl</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>a</mi></mfenced></mrow> |
<mrow><mo>∇<!--NABLA--></mo><mo>×<!--MULTIPLICATION SIGN--></mo><mi>a</mi></mrow> |
The functions defining the coordinates may be defined implicitly as expressions defined in terms of the coordinate names, in which case the coordinate names must be provided as bound variables.
<laplacian/>
Class | unary-veccalc |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | Laplacian |
The laplacian
element represents the Laplacian operator of
vector calculus. The Laplacian takes a single argument which is a
vector of scalar-valued functions representing a vector-valued
function, and returns a vector of functions.
Content MathML
<apply><laplacian/><ci type="vector">E</ci></apply> |
Sample Presentation
<mrow> <msup><mo>∇<!--NABLA--></mo><mn>2</mn></msup> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>E</mi></mfenced> </mrow> |
The functions defining the coordinates may be defined implicitly as expressions defined in terms of the coordinate names, in which case the coordinate names must be provided as bound variables.
Content MathML
<apply><laplacian/> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <bvar><ci>z</ci></bvar> <apply><ci>f</ci><ci>x</ci><ci>y</ci></apply> </apply> |
Sample Presentation
<mrow> <msup><mo>∇<!--NABLA--></mo><mn>2</mn></msup> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mrow> <mo>(</mo> <mfenced><mi>x</mi><mi>y</mi><mi>z</mi></mfenced> <mo>↦<!--RIGHTWARDS ARROW FROM BAR--></mo> <mrow> <mi>f</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>x</mi><mi>y</mi></mfenced> </mrow> <mo>)</mo> </mrow> </mrow> |
<set>
Class | nary-setlist-constructor |
---|---|
Attributes | CommonAtt, DefEncAtt, type? |
type Attribute Values
|
"set" | "multiset" | text |
Content | ContExp* |
Qualifiers | BvarQ,DomainQ |
OM Symbols | set, multiset |
The set
represents a function which constructs
mathematical sets from its arguments. It is an n-ary function. The
members of the set to be constructed may be given explicitly as
child elements of the constructor, or specified by rule as described
in Section 4.3.1.1 Container Markup for Constructor Symbols. There is no implied ordering to
the elements of a set.
Content MathML
<set> <ci>a</ci><ci>b</ci><ci>c</ci> </set> |
Sample Presentation
<mrow> <mo>{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>}</mo> </mrow> |
In general, a set can be constructed by providing a function and a domain of application. The elements of the set correspond to the values obtained by evaluating the function at the points of the domain.
Content MathML
<set> <bvar><ci>x</ci></bvar> <condition> <apply><lt/><ci>x</ci><cn>5</cn></apply> </condition> <ci>x</ci> </set> |
Sample Presentation
<mrow> <mo>{</mo> <mi>x</mi> <mo>|</mo> <mrow><mi>x</mi><mo><</mo><mn>5</mn></mrow> <mo>}</mo> </mrow> |
Content MathML
<set> <bvar><ci type="set">S</ci></bvar> <condition> <apply><in/><ci>S</ci><ci type="list">T</ci></apply> </condition> <ci>S</ci> </set> |
Sample Presentation
<mrow> <mo>{</mo> <mi>S</mi> <mo>|</mo> <mrow><mi>S</mi><mo>∈<!--ELEMENT OF--></mo><mi>T</mi></mrow> <mo>}</mo> </mrow> |
Content MathML
<set> <bvar><ci> x </ci></bvar> <condition> <apply><and/> <apply><lt/><ci>x</ci><cn>5</cn></apply> <apply><in/><ci>x</ci><naturalnumbers/></apply> </apply> </condition> <ci>x</ci> </set> |
Sample Presentation
<mrow> <mo>{</mo> <mi>x</mi> <mo>|</mo> <mrow> <mrow><mo>(</mo><mi>x</mi><mo><</mo><mn>5</mn><mo>)</mo></mrow> <mo>∧<!--LOGICAL AND--></mo> <mrow> <mi>x</mi><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo>}</mo> </mrow> |
<list>
Class | nary-setlist-constructor |
---|---|
Attributes | CommonAtt, DefEncAtt, order |
order Attribute Values
|
"numeric" | "lexicographic" |
Content | ContExp* |
Qualifiers | BvarQ,DomainQ |
OM Symbols | interval_cc, list |
The list
elements represents the n-ary function which
constructs a list from its arguments. Lists differ from sets in that
there is an explicit order to the elements.
The list entries and order may be given explicitly.
Content MathML
<list> <ci>a</ci><ci>b</ci><ci>c</ci> </list> |
Sample Presentation
<mrow> <mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo> </mrow> |
In general a list can be constructed by providing a function and
a domain of application. The elements of the list correspond to the
values obtained by evaluating the function at the points of the
domain. When this method is used, the ordering of the list elements
may not be clear, so the kind of ordering may be specified by the
order
attribute. Two orders are supported: lexicographic
and numeric.
Content MathML
<list order="numeric"> <bvar><ci>x</ci></bvar> <condition> <apply><lt/><ci>x</ci><cn>5</cn></apply> </condition> </list> |
Sample Presentation
<mrow> <mo>(</mo> <mi>x</mi> <mo>|</mo> <mrow><mi>x</mi><mo><</mo><mn>5</mn></mrow> <mo>)</mo> </mrow> |
<union/>
Class | nary-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | union |
The union
element is used to denote the n-ary union of sets. It takes sets as arguments,
and denotes the set that contains all the elements that occur in any
of them.
Arguments may be explicitly specified.
Content MathML
<apply><union/><ci>A</ci><ci>B</ci></apply> |
Sample Presentation
<mrow><mi>A</mi><mo>∪<!--UNION--></mo><mi>B</mi></mrow> |
Arguments may also be specified using qualifier elements as described in Section 4.3.4.1 N-ary Operators (classes nary-arith, nary-functional, nary-logical, nary-linalg, nary-set, nary-constructor). operator element can be used as a binding operator to construct the union over a collection of sets.
Content MathML
<apply><union/> <bvar><ci type="set">S</ci></bvar> <domainofapplication> <ci type="list">L</ci> </domainofapplication> <ci type="set"> S</ci> </apply> |
Sample Presentation
<mrow><munder><mo>⋃<!--N-ARY UNION--></mo><mi>L</mi></munder><mi>S</mi></mrow> |
<intersect/>
Class | nary-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | intersect |
The intersect
element is used to denote the n-ary
intersection of sets. It takes sets as arguments, and denotes the
set that contains all the elements that occur in all of them. Its arguments may be explicitly specified in the
enclosing apply
element, or specified using qualifier
elements as described in Section 4.3.4.1 N-ary Operators (classes nary-arith, nary-functional, nary-logical,
nary-linalg, nary-set, nary-constructor).
Content MathML
<apply><intersect/> <ci type="set"> A </ci> <ci type="set"> B </ci> </apply> |
Sample Presentation
<mrow><mi>A</mi><mo>∩<!--INTERSECTION--></mo><mi>B</mi></mrow> |
Content MathML
<apply><intersect/> <bvar><ci type="set">S</ci></bvar> <domainofapplication><ci type="list">L</ci></domainofapplication> <ci type="set"> S </ci> </apply> |
Sample Presentation
<mrow><munder><mo>⋂<!--N-ARY INTERSECTION--></mo><mi>L</mi></munder><mi>S</mi></mrow> |
<in/>
Class | binary-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | in |
The in
element represents the set inclusion relation.
It has two arguments, an element and a set. It is used to denote
that the element is in the given set.
Content MathML
<apply><in/><ci>a</ci><ci type="set">A</ci></apply> |
Sample Presentation
<mrow><mi>a</mi><mo>∈<!--ELEMENT OF--></mo><mi>A</mi></mrow> |
When translating to Strict Content Markup, if the type
has value "multiset", then the in symbol from multiset1 should
be used instead.
<notin/>
Class | binary-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | notin |
The notin
represents the negated set inclusion
relation. It has two arguments, an element and a set. It is
used to denote that the element is not in the given set.
Content MathML
<apply><notin/><ci>a</ci><ci type="set">A</ci></apply> |
Sample Presentation
<mrow><mi>a</mi><mo>∉<!--NOT AN ELEMENT OF--></mo><mi>A</mi></mrow> |
When translating to Strict Content Markup, if the type
has value "multiset", then
the in symbol from multiset1 should be used instead.
<subset/>
Class | nary-set-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | subset |
The subset
element represents the subset relation. It is used to denote that the
first argument is a subset of the second. As described in Section 4.3.4.3 N-ary Relations (classes nary-reln, nary-set-reln), it may also be used as an n-ary operator to express
that each argument is a subset of its predecessor.
Content MathML
<apply><subset/> <ci type="set">A</ci> <ci type="set">B</ci> </apply> |
Sample Presentation
<mrow><mi>A</mi><mo>⊆<!--SUBSET OF OR EQUAL TO--></mo><mi>B</mi></mrow> |
<prsubset/>
Class | nary-set-reln |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | prsubset |
The prsubset
element represents the proper subset
relation, i.e. that the first argument is a proper subset of the
second. As described in Section 4.3.4.3 N-ary Relations (classes nary-reln, nary-set-reln), it may
also be used as an n-ary operator to express that each argument is a
proper subset of its predecessor.
Content MathML
<apply><prsubset/> <ci type="set">A</ci> <ci type="set">B</ci> </apply> |
Sample Presentation
<mrow><mi>A</mi><mo>⊂<!--SUBSET OF--></mo><mi>B</mi></mrow> |
<notsubset/>
Class | binary-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | notsubset |
The notsubset
element represents the negated subset
relation. It is used to denote that the first argument is not a subset of the
second.
Content MathML
<apply><notsubset/> <ci type="set">A</ci> <ci type="set">B</ci> </apply> |
Sample Presentation
<mrow><mi>A</mi><mo>⊈<!--NEITHER A SUBSET OF NOR EQUAL TO--></mo><mi>B</mi></mrow> |
When translating to Strict Content Markup, if the type
has value "multiset", then
the in symbol from multiset1 should be used instead.
<notprsubset/>
Class | binary-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | notprsubset |
The notprsubset
element represents the negated proper
subset relation. It is used to denote that the first argument is not
a proper subset of the second.
Content MathML
<apply><notprsubset/> <ci type="set">A</ci> <ci type="set">B</ci> </apply> |
Sample Presentation
<mrow><mi>A</mi><mo>⊄<!--NOT A SUBSET OF--></mo><mi>B</mi></mrow> |
When translating to Strict Content Markup, if the type
has value "multiset", then
the in symbol from multiset1 should be used instead.
<setdiff/>
Class | binary-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | setdiff, setdiff |
The setdiff
element represents set difference
operator. It takes two sets as arguments, and denotes the set that
contains all the elements that occur in the first set, but not in
the second.
Content MathML
<apply><setdiff/> <ci type="set">A</ci> <ci type="set">B</ci> </apply> |
Sample Presentation
<mrow><mi>A</mi><mo>∖<!--SET MINUS--></mo><mi>B</mi></mrow> |
When translating to Strict Content Markup, if the type
has value "multiset", then
the in symbol from multiset1 should be used instead.
<card/>
Class | unary-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | size, size |
The card
element represents the cardinality function,
which takes a set argument and returns its cardinality, i.e. the
number of elements in the set. The cardinality of a set is a
non-negative integer, or an infinite cardinal number.
Content MathML
<apply><eq/> <apply><card/><ci>A</ci></apply> <cn>5</cn> </apply> |
Sample Presentation
<mrow> <mrow><mo>|</mo><mi>A</mi><mo>|</mo></mrow> <mo>=</mo> <mn>5</mn> </mrow> |
When translating to Strict Content Markup, if the type
has value "multiset", then the size symbol from multiset1 should be used
instead.
<cartesianproduct/>
Class | nary-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | cartesian_product |
The cartesianproduct
element is used to represents the
Cartesian product operator. It takes sets as arguments, which may be
explicitly specified in the enclosing apply
element, or
specified using qualifier elements as described in Section 4.3.4.1 N-ary Operators (classes nary-arith, nary-functional, nary-logical,
nary-linalg, nary-set, nary-constructor).
Content MathML
<apply><cartesianproduct/><ci>A</ci><ci>B</ci></apply> |
Sample Presentation
<mrow><mi>A</mi><mo>×<!--MULTIPLICATION SIGN--></mo><mi>B</mi></mrow> |
<sum/>
Class | sum |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | sum |
The sum
element represents the n-ary addition operator.
The terms of the sum are normally specified by rule through the use of
qualifiers. While it can be used with an explicit list of
arguments, this is strongly discouraged, and the plus
operator should be used instead in such situations.
The sum
operator may be used either with or without
explicit bound variables. When a bound variable is used, the
sum
element is followed by one or more bvar
elements giving the index variables, followed by qualifiers giving
the domain for the index variables. The final child in the enclosing
apply
is then an expression in the bound variables, and the
terms of the sum are obtained by evaluating this expression at each
point of the domain of the index variables. Depending on the
structure of the domain, the domain of summation is often given
by using uplimit
and lowlimit
to specify upper and
lower limits for the sum.
When no bound variables are explicitly given, the final child of
the enclosing apply
element must be a function, and the
terms of the sum are obtained by evaluating the function at
each point of the domain specified by qualifiers.
Content MathML
<apply><sum/> <bvar><ci>x</ci></bvar> <lowlimit><ci>a</ci></lowlimit> <uplimit><ci>b</ci></uplimit> <apply><ci>f</ci><ci>x</ci></apply> </apply> |
Sample Presentation
<mrow> <munderover> <mo>∑<!--N-ARY SUMMATION--></mo> <mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow> <mi>b</mi> </munderover> <mrow><mi>f</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi></mfenced></mrow> </mrow> |
Content MathML
<apply><sum/> <bvar><ci>x</ci></bvar> <condition> <apply><in/><ci>x</ci><ci type="set">B</ci></apply> </condition> <apply><ci type="function">f</ci><ci>x</ci></apply> </apply> |
Sample Presentation
<mrow> <munder> <mo>∑<!--N-ARY SUMMATION--></mo> <mrow><mi>x</mi><mo>∈<!--ELEMENT OF--></mo><mi>B</mi></mrow> </munder> <mrow><mi>f</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi></mfenced></mrow> </mrow> |
Content MathML
<apply><sum/> <domainofapplication> <ci type="set">B</ci> </domainofapplication> <ci type="function">f</ci> </apply> |
Sample Presentation
<mrow><munder><mo>∑<!--N-ARY SUMMATION--></mo><mi>B</mi></munder><mi>f</mi></mrow> |
Mapping to Strict Content MathML
When no explicit bound variables are used, no special rules are
required to rewrite sums as Strict Content beyond the generic rules
for rewriting expressions using qualifiers. However, when bound
variables are used, it is necessary to introduce a lambda
construction to rewrite the expression in the bound variables as a
function.
Content MathML
<apply><sum/> <bvar><ci>i</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><cn>100</cn></uplimit> <apply><power/><ci>x</ci><ci>i</ci></apply> </apply> |
Strict Content MathML equivalent
<apply><csymbol cd="arith1">sum</csymbol> <apply><csymbol cd="interval1">integer_interval</csymbol> <cn>0</cn> <cn>100</cn> </apply> <bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>i</ci></bvar> <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><ci>i</ci></apply> </bind> </apply> |
<product/>
Class | product |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | product |
The product
element represents the n-ary multiplication operator.
The terms of the product are normally specified by rule through the use of
qualifiers. While it can be used with an explicit list of
arguments, this is strongly discouraged, and the times
operator should be used instead in such situations.
The product
operator may be used either with or without
explicit bound variables. When a bound variable is used, the
product
element is followed by one or more bvar
elements giving the index variables, followed by qualifiers giving
the domain for the index variables. The final child in the enclosing
apply
is then an expression in the bound variables, and the
terms of the product are obtained by evaluating this expression at
each point of the domain. Depending on the structure of the domain,
it is commonly given using uplimit
and lowlimit
qualifiers.
When no bound variables are explicitly given, the final child of
the enclosing apply
element must be a function, and the
terms of the product are obtained by evaluating the function
at each point of the domain specified by qualifiers.
Content MathML
<apply><product/> <bvar><ci>x</ci></bvar> <lowlimit><ci>a</ci></lowlimit> <uplimit><ci>b</ci></uplimit> <apply><ci type="function">f</ci> <ci>x</ci> </apply> </apply> |
Sample Presentation
<mrow> <munderover> <mo>∏<!--N-ARY PRODUCT--></mo> <mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow> <mi>b</mi> </munderover> <mrow><mi>f</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi></mfenced></mrow> </mrow> |
Content MathML
<apply><product/> <bvar><ci>x</ci></bvar> <condition> <apply><in/> <ci>x</ci> <ci type="set">B</ci> </apply> </condition> <apply><ci>f</ci><ci>x</ci></apply> </apply> |
Sample Presentation
<mrow> <munder> <mo>∏<!--N-ARY PRODUCT--></mo> <mrow><mi>x</mi><mo>∈<!--ELEMENT OF--></mo><mi>B</mi></mrow> </munder> <mrow><mi>f</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi></mfenced></mrow> </mrow> |
Mapping to Strict Content MathML
When no explicit bound variables are used, no special rules are
required to rewrite products as Strict Content beyond the generic rules
for rewriting expressions using qualifiers. However, when bound
variables are used, it is necessary to introduce a lambda
construction to rewrite the expression in the bound variables as a
function.
Content MathML
<apply><product/> <bvar><ci>i</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><cn>100</cn></uplimit> <apply><power/><ci>x</ci><ci>i</ci></apply> </apply> |
Strict Content MathML equivalent
<apply><csymbol cd="arith1">product</csymbol> <apply><csymbol cd="interval1">integer_interval</csymbol> <cn>0</cn> <cn>100</cn> </apply> <bind><csymbol cd="fns1">lambda</csymbol> <bvar><ci>i</ci></bvar> <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><ci>i</ci></apply> </bind> </apply> |
<limit/>
Class | limit |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | lowlimit, condition |
OM Symbols | limit, both_sides, above, below, null |
The limit
element represents the operation of taking a limit of a
sequence. The limit point is expressed by specifying a lowlimit
and a
bvar
, or by specifying a condition
on one or more bound variables.
Content MathML
<apply><limit/> <bvar><ci>x</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <apply><sin/><ci>x</ci></apply> </apply> |
Sample Presentation
<mrow> <munder> <mi>lim</mi> <mrow><mi>x</mi><mo>→<!--RIGHTWARDS ARROW--></mo><mn>0</mn></mrow> </munder> <mrow><mi>sin</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow> </mrow> |
Content MathML
<apply><limit/> <bvar><ci>x</ci></bvar> <condition> <apply><tendsto/><ci>x</ci><cn>0</cn></apply> </condition> <apply><sin/><ci>x</ci></apply> </apply> |
Sample Presentation
<mrow> <munder> <mi>lim</mi> <mrow><mi>x</mi><mo>→<!--RIGHTWARDS ARROW--></mo><mn>0</mn></mrow> </munder> <mrow><mi>sin</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow> </mrow> |
Content MathML
<apply><limit/> <bvar><ci>x</ci></bvar> <condition> <apply><tendsto type="above"/><ci>x</ci><ci>a</ci></apply> </condition> <apply><sin/><ci>x</ci></apply> </apply> |
Sample Presentation
<mrow> <munder> <mi>lim</mi> <mrow><mi>x</mi><mo>→<!--RIGHTWARDS ARROW--></mo><msup><mi>a</mi><mo>+</mo></msup></mrow> </munder> <mrow><mi>sin</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow> </mrow> |
The direction from which a limiting value is approached is given as an argument
limit in Strict Content MathML, which supplies the
direction specifier symbols both_sides, above, and below for this
purpose. The first correspond to the values "all", "above",
and "below" of the type
attribute of the tendsto
element below. The null symbol corresponds to the case
where no type
attribute is present. We translate
<apply><limit/> <bvar></bvar> <condition> <apply><tendsto/> </apply> </condition> </apply> |
Strict Content MathML equivalent
<apply><csymbol cd="limit1">limit</csymbol><csymbol cd="limit1"> </csymbol> <bind><csymbol cd="fns1">lambda</csymbol> <bvar> </bvar> </bind> </apply> |
where
is an
arbitrary expression involving the bound variable(s), and the choice of
symbol, null depends on the
type
attribute of the tendsto
element as
described above.
<tendsto/>
Class | binary-reln |
---|---|
Attributes | CommonAtt, DefEncAtt, type? |
type Attribute Values
|
string |
Content | Empty |
OM Symbols | limit |
The tendsto
element is used to express the relation that
a quantity is tending to a specified value. While this is used
primarily as part of the statement of a mathematical limit, it
exists as a construct on its own to allow one to capture
mathematical statements such as "As x tends to y," and to provide a
building block to construct more general kinds of limits.
The tendsto
element takes the attributes type
to set the
direction from which the limiting value is approached.
Content MathML
<apply><tendsto type="above"/> <apply><power/><ci>x</ci><cn>2</cn></apply> <apply><power/><ci>a</ci><cn>2</cn></apply> </apply> |
Sample Presentation
<mrow> <msup><mi>x</mi><mn>2</mn></msup> <mo>→<!--RIGHTWARDS ARROW--></mo> <msup><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo></msup> </mrow> |
Content MathML
<apply><tendsto/> <vector><ci>x</ci><ci>y</ci></vector> <vector> <apply><ci type="function">f</ci><ci>x</ci><ci>y</ci></apply> <apply><ci type="function">g</ci><ci>x</ci><ci>y</ci></apply> </vector> </apply> |
Sample Presentation
<mfenced><mtable> <mtr><mtd><mi>x</mi></mtd></mtr> <mtr><mtd><mi>y</mi></mtd></mtr> </mtable></mfenced> <mo>→<!--RIGHTWARDS ARROW--></mo> <mfenced><mtable> <mtr><mtd> <mi>f</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi><mi>y</mi></mfenced> </mtd></mtr> <mtr><mtd> <mi>g</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>x</mi><mi>y</mi></mfenced> </mtd></mtr> </mtable></mfenced> |
Mapping to Strict Content MathML
The usage of tendsto
to qualify a limit is formally
defined by writing the expression in Strict Content MathML via the
rule Rewrite: limits condition. The meanings of other more
idiomatic uses of tendsto
are not formally defined by this
specification. When rewriting these cases to Strict Content MathML,
tendsto
should be rewritten to an annotated identifier as
shown below.
<sin/>
,
<cos/>
,
<tan/>
,
<sec/>
,
<csc/>
,
<cot/>
Class | unary-elementary |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | sin |
These operator elements denote the standard trigonometric functions. Since their standard interpretations are widely known, they are discussed as a group.
Content MathML
<apply><sin/><ci>x</ci></apply> |
Sample Presentation
<mrow><mi>sin</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow> |
Content MathML
<apply><sin/> <apply><plus/> <apply><cos/><ci>x</ci></apply> <apply><power/><ci>x</ci><cn>3</cn></apply> </apply> </apply> |
Sample Presentation
<mrow> <mi>sin</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mrow> <mo>(</mo> <mrow><mi>cos</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow> <mo>+</mo> <msup><mi>x</mi><mn>3</mn></msup> <mo>)</mo> </mrow> </mrow> |
<arcsin/>
,
<arccos/>
,
<arctan/>
,
<arcsec/>
,
<arccsc/>
,
<arccot/>
Class | unary-elementary |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | arcsin |
These operator elements denote the inverses of standard trigonometric functions. Differing definitions are in use so for maximum interoperability applications evaluating such expressions should follow the definitions in [Abramowitz1977].
Content MathML
<apply><arcsin/><ci>x</ci></apply> |
Sample Presentations
<mrow> <mi>arcsin</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mi>x</mi> </mrow> |
<mrow> <msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mi>x</mi> </mrow> |
<sinh/>
,
<cosh/>
,
<tanh/>
,
<sech/>
,
<csch/>
,
<coth/>
Class | unary-elementary |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | sinh |
These operator elements denote the standard hyperbolic functions. Since their standard interpretations are widely known, they are discussed as a group.
Content MathML
<apply><sinh/><ci>x</ci></apply> |
Sample Presentation
<mrow><mi>sinh</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow> |
<arcsinh/>
,
<arccosh/>
,
<arctanh/>
,
<arcsech/>
,
<arccsch/>
,
<arccoth/>
Class | unary-elementary |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | arcsinh |
These operator elements denote the inverses of standard hyperbolic functions. Differing definitions are in use so for maximum interoperability applications evaluating such expressions should follow the definitions in [Abramowitz1977].
Content MathML
<apply><arcsinh/><ci>x</ci></apply> |
Sample Presentations
<mrow> <mi>arcsinh</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mi>x</mi> </mrow> |
<mrow> <msup><mi>sinh</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mi>x</mi> </mrow> |
<exp/>
Class | unary-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | exp |
The exp
element represents the exponentiation function
associated with the inverse of the ln function. It takes one
argument.
Content MathML
<apply><exp/><ci>x</ci></apply> |
Sample Presentation
<msup><mi>e</mi><mi>x</mi></msup> |
<ln/>
Class | unary-functional |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | ln |
The ln
element represents the natural logarithm function.
Content MathML
<apply><ln/><ci>a</ci></apply> |
Sample Presentation
<mrow><mi>ln</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>a</mi></mrow> |
<log/>
, <logbase>
Class | unary-functional |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | logbase |
OM Symbols | log |
The log
elements represents the logarithm function
relative to a given base. When present, the logbase
qualifier specifies the base. Otherwise, the base is assumed to be 10.
apply
.
Content MathML
<apply><log/> <logbase><cn>3</cn></logbase> <ci>x</ci> </apply> |
Sample Presentation
<mrow><msub><mi>log</mi><mn>3</mn></msub><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow> |
Content MathML
<apply><log/><ci>x</ci></apply> |
Sample Presentation
<mrow><mi>log</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>x</mi></mrow> |
Mapping to Strict Content MathML
When mapping log
to Strict Content, one uses the
log symbol denoting the function
that returns the log of its second argument with respect to the base
specified by the first argument. When logbase
is present, it
determines the base. Otherwise, the default base of 10 must be
explicitly provided in Strict markup. See the following example.
<apply><plus/> <apply> <log/> <logbase><cn>2</cn></logbase> <ci>x</ci> </apply> <apply> <log/> <ci>y</ci> </apply> </apply> |
Strict Content MathML equivalent:
<apply> <csymbol cd="arith1">plus</csymbol> <apply> <csymbol cd="transc1">log</csymbol> <cn>2</cn> <ci>x</ci> </apply> <apply> <csymbol cd="transc1">log</csymbol> <cn>10</cn> <ci>y</ci> </apply> </apply> |
<mean/>
Class | nary-stats |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | mean, mean |
The mean
element represents the function returning arithmetic mean or average of a
data set or random variable.
Content MathML
<apply><mean/> <cn>3</cn><cn>4</cn><cn>3</cn><cn>7</cn><cn>4</cn> </apply> |
Sample Presentation
<mrow> <mo>⟨<!--MATHEMATICAL LEFT ANGLE BRACKET--></mo> <mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>3</mn> <mo>,</mo><mn>7</mn><mo>,</mo><mn>4</mn> <mo>⟩<!--MATHEMATICAL RIGHT ANGLE BRACKET--></mo> </mrow> |
Content MathML
<apply><mean/><ci>X</ci></apply> |
Sample Presentation
<mrow><mo>⟨<!--MATHEMATICAL LEFT ANGLE BRACKET--></mo><mi>X</mi><mo>⟩<!--MATHEMATICAL RIGHT ANGLE BRACKET--></mo></mrow> |
<mover><mi>X</mi><mo>¯<!--MACRON--></mo></mover> |
Mapping to Strict Markup
When the mean
element is applied to an explicit list of arguments, the
translation to Strict Content markup is direct, using the mean symbol from the s_data1 content dictionary, as described in
Rewrite: element. When it is applied to a distribution, then the
mean symbol from the s_dist1 content
dictionary should be used. In the case with qualifiers use Rewrite: n-ary domainofapplication
with the same caveat.
<sdev/>
Class | nary-stats |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | sdev, sdev |
The sdev
element is used to denote the standard deviation
function for a data set or random variable. Standard deviation is a
statistical measure of dispersion given by the square root of the
variance.
Content MathML
<apply><sdev/> <cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn> </apply> |
Sample Presentation
<mrow> <mo>σ<!--GREEK SMALL LETTER SIGMA--></mo> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mn>3</mn><mn>4</mn><mn>2</mn><mn>2</mn></mfenced> </mrow> |
Content MathML
<apply><sdev/> <ci type="discrete_random_variable">X</ci> </apply> |
Sample Presentation
<mrow><mo>σ<!--GREEK SMALL LETTER SIGMA--></mo><mo>⁡<!--FUNCTION APPLICATION--></mo><mfenced><mi>X</mi></mfenced></mrow> |
Mapping to Strict Markup
When the sdev
element is applied to an explicit list of arguments, the
translation to Strict Content markup is direct, using the sdev
symbol from the s_data1 content dictionary, as described in
Rewrite: element. When it is applied to a distribution, then the
sdev symbol from the s_dist1 content
dictionary should be used. In the case with qualifiers use
Rewrite: n-ary domainofapplication with the same caveat.
<variance/>
Class | nary-stats |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ,DomainQ |
OM Symbols | variance, variance |
The variance
element represents the variance of a data set
or random variable. Variance is a statistical measure of dispersion,
averaging the squares of the deviations of possible values from their
mean.
Content MathML
<apply><variance/> <cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn> </apply> |
Sample Presentation
<mrow> <msup> <mo>σ<!--GREEK SMALL LETTER SIGMA--></mo> <mn>2</mn> </msup> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mn>3</mn><mn>4</mn><mn>2</mn><mn>2</mn></mfenced> </mrow> |
Content MathML
<apply><variance/> <ci type="discrete_random_variable"> X</ci> </apply> |
Sample Presentation
<mrow> <msup><mo>σ<!--GREEK SMALL LETTER SIGMA--></mo><mn>2</mn></msup> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>X</mi></mfenced> </mrow> |
Mapping to Strict Markup
When the variance
element is applied to an explicit list of arguments, the
translation to Strict Content markup is direct, using the variance
symbol from the s_data1 content dictionary, as described in
Rewrite: element. When it is applied to a distribution, then the
variance symbol from the s_dist1 content
dictionary should be used. In the case with qualifiers use Rewrite: n-ary domainofapplication
with the same caveat.
<median/>
Class | nary-stats |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ, DomainQ |
OM Symbols | median |
The median
element represents an operator returning the
median of its arguments. The median is a number separating the lower
half of the sample values from the upper half.
Content MathML
<apply><median/> <cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn> </apply> |
Sample Presentation
<mrow> <mi>median</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mn>3</mn><mn>4</mn><mn>2</mn><mn>2</mn></mfenced> </mrow> |
Mapping to Strict Markup
When the median
element is applied to an explicit list of arguments, the
translation to Strict Content markup is direct, using the median
symbol from the s_data1 content dictionary, as described in
Rewrite: element.
<mode/>
Class | nary-stats |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | BvarQ, DomainQ |
OM Symbols | mode |
The mode
element is used to denote the mode of its arguments. The mode is
the value which occurs with the greatest frequency.
Content MathML
<apply><mode/> <cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn> </apply> |
Sample Presentation
<mrow> <mi>mode</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mn>3</mn><mn>4</mn><mn>2</mn><mn>2</mn></mfenced> </mrow> |
Mapping to Strict Markup
When the mode
element is applied to an explicit list of arguments, the
translation to Strict Content markup is direct, using the mode
symbol from the s_data1 content dictionary, as described in
Rewrite: element.
<moment/>
, <momentabout>
Class | unary-functional |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
Qualifiers | degree, momentabout |
OM Symbols | moment, moment |
The moment
element is used to denote the ith moment of a set of data set or
random variable. The moment
function accepts the degree
and
momentabout
qualifiers. If present, the degree
schema denotes the order of
the moment. Otherwise, the moment is assumed to be the first order moment. When used with
moment
, the degree
schema is expected to contain a
single child. If present, the momentabout
schema denotes the
point about which the moment is taken. Otherwise, the moment is
assumed to be the moment about zero.
Content MathML
<apply><moment/> <degree><cn>3</cn></degree> <momentabout><mean/></momentabout> <cn>6</cn><cn>4</cn><cn>2</cn><cn>2</cn><cn>5</cn> </apply> |
Sample Presentation
<msub> <mrow> <mo>⟨<!--MATHEMATICAL LEFT ANGLE BRACKET--></mo> <msup> <mfenced><mn>6</mn><mn>4</mn><mn>2</mn><mn>2</mn><mn>5</mn></mfenced> <mn>3</mn> </msup> <mo>⟩<!--MATHEMATICAL RIGHT ANGLE BRACKET--></mo> </mrow> <mi>mean</mi> </msub> |
Content MathML
<apply><moment/> <degree><cn>3</cn></degree> <momentabout><ci>p</ci></momentabout> <ci>X</ci> </apply> |
Sample Presentation
<msub> <mrow> <mo>⟨<!--MATHEMATICAL LEFT ANGLE BRACKET--></mo><msup><mi>X</mi><mn>3</mn></msup><mo>⟩<!--MATHEMATICAL RIGHT ANGLE BRACKET--></mo> </mrow> <mi>p</mi> </msub> |
Mapping to Strict Markup
When rewriting to Strict Markup, the moment symbol from the s_data1 content
dictionary is used when the moment
element is applied
to an explicit list of arguments. When it is applied to a distribution, then the
moment symbol from the s_dist1 content
dictionary should be used. Both operators take
the degree as the first argument, the point as the second, followed by
the data set or random variable respectively.
<apply><moment/> <degree><cn>3</cn></degree> <momentabout><ci>p</ci></momentabout> <ci>X</ci> </apply> |
Strict Content MathML equivalent
<apply><csymbol cd="s_dist1">moment</csymbol> <cn>3</cn> <ci>p</ci> <ci>X</ci> </apply> |
<vector>
Class | nary-constructor |
---|---|
Attributes | CommonAtt, DefEncAtt |
Qualifiers | BvarQ, DomainQ |
Content | ContExp* |
OM Symbol | vector |
A vector is an ordered n-tuple of values representing an element of an n-dimensional vector space.
For purposes of interaction with matrices and matrix multiplication, vectors are regarded as equivalent to a matrix consisting of a single column, and the transpose of a vector as a matrix consisting of a single row.
The components of a vector
may be given explicitly as
child elements, or specified by rule as described in Section 4.3.1.1 Container Markup for Constructor Symbols.
Content MathML
<vector> <apply><plus/><ci>x</ci><ci>y</ci></apply> <cn>3</cn> <cn>7</cn> </vector> |
Sample Presentation
<mrow> <mo>(</mo> <mtable> <mtr><mtd><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mtd></mtr> <mtr><mtd><mn>3</mn></mtd></mtr> <mtr><mtd><mn>7</mn></mtd></mtr> </mtable> <mo>)</mo> </mrow> |
<mfenced> <mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow> <mn>3</mn> <mn>7</mn> </mfenced> |
<matrix>
Class | nary-constructor |
---|---|
Attributes | CommonAtt, DefEncAtt |
Qualifiers | BvarQ, DomainQ |
Content | ContExp* |
OM Symbol | matrix |
A matrix is regarded as made up of matrix rows, each of which can be thought of as a special type of vector.
Note that the behavior of the matrix
and matrixrow
elements is
substantially different from the mtable
and mtr
presentation
elements.
The matrix
element is a constructor
element, so the entries may be given explicitly as child elements,
or specified by rule as described in Section 4.3.1.1 Container Markup for Constructor Symbols. In the latter case, the
entries are specified by providing a function and a 2-dimensional
domain of application. The entries of the matrix correspond to
the values obtained by evaluating the function at the points of
the domain.
Content MathML
<matrix> <bvar><ci type="integer">i</ci></bvar> <bvar><ci type="integer">j</ci></bvar> <condition> <apply><and/> <apply><in/> <ci>i</ci> <interval><ci>1</ci><ci>5</ci></interval> </apply> <apply><in/> <ci>j</ci> <interval><ci>5</ci><ci>9</ci></interval> </apply> </apply> </condition> <apply><power/><ci>i</ci><ci>j</ci></apply> </matrix> |
Sample Presentation
<mrow> <mo>[</mo> <msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub> <mo>|</mo> <mrow> <msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub> <mo>=</mo> <msup><mi>i</mi><mi>j</mi></msup> </mrow> <mo>;</mo> <mrow> <mrow> <mi>i</mi> <mo>∈<!--ELEMENT OF--></mo> <mfenced open="[" close="]"><mi>1</mi><mi>5</mi></mfenced> </mrow> <mo>∧<!--LOGICAL AND--></mo> <mrow> <mi>j</mi> <mo>∈<!--ELEMENT OF--></mo> <mfenced open="[" close="]"><mi>5</mi><mi>9</mi></mfenced> </mrow> </mrow> <mo>]</mo> </mrow> |
<matrixrow>
Class | nary-constructor |
---|---|
Attributes | CommonAtt, DefEncAtt |
Qualifiers | BvarQ, DomainQ |
Content | ContExp* |
OM Symbol | matrixrow |
This element is an n-ary constructor used to represent rows of matrices.
Matrix rows are not directly rendered by themselves outside of the context of a matrix.
<determinant/>
Class | unary-linalg |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | determinant |
This element is used for the unary function which returns the determinant of its argument, which should be a square matrix.
Content MathML
<apply><determinant/> <ci type="matrix">A</ci> </apply> |
Sample Presentation
<mrow><mi>det</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>A</mi></mrow> |
<transpose/>
Class | unary-linalg |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | transpose |
This element represents a unary function that signifies the transpose of the given matrix or vector.
Content MathML
<apply><transpose/> <ci type="matrix">A</ci> </apply> |
Sample Presentation
<msup><mi>A</mi><mi>T</mi></msup> |
<selector/>
Class | nary-linalg |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | vector_selector, matrix_selector |
The selector
element is the operator for indexing into vectors, matrices
and lists. It accepts one or more arguments. The first argument identifies the vector,
matrix or list from which the selection is taking place, and the second and subsequent
arguments, if any, indicate the kind of selection taking place.
When selector
is used with a single argument, it should be interpreted as
giving the sequence of all elements in the list, vector or matrix given. The ordering
of elements in the sequence for a matrix is understood to be first by column, then by
row; so the resulting list is of matrix rows given entry by entry.
That is, for a matrix (ai,j), where the indices denote row
and column, respectively, the ordering would be a1,1,
a1,2, ... a2,1, a2,2
... etc.
When two arguments are given, and the first is a vector or list, the second argument specifies the index of an entry in the list or vector. If the first argument is a matrix then the second argument specifies the index of a matrix row.
When three arguments are given, the last one is ignored for a list or vector, and in the case of a matrix, the second and third arguments specify the row and column indices of the selected element.
Content MathML
<apply><selector/><ci type="vector">V</ci><cn>1</cn></apply> |
Sample Presentation
<msub><mi>V</mi><mn>1</mn></msub> |
Content MathML
<apply><eq/> <apply><selector/> <matrix> <matrixrow><cn>1</cn><cn>2</cn></matrixrow> <matrixrow><cn>3</cn><cn>4</cn></matrixrow> </matrix> <cn>1</cn> </apply> <matrix> <matrixrow><cn>1</cn><cn>2</cn></matrixrow> </matrix> </apply> |
Sample Presentation
<mrow> <msub> <mrow> <mo>(</mo> <mtable> <mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr> <mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr> </mtable> <mo>)</mo> </mrow> <mn>1</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable> <mo>)</mo> </mrow> </mrow> |
<vectorproduct/>
Class | binary-linalg |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | vectorproduct |
This element represents the vector product. It takes two three-dimensional vector arguments and represents as value a three-dimensional vector.
Content MathML
<apply><eq/> <apply><vectorproduct/> <ci type="vector"> A </ci> <ci type="vector"> B </ci> </apply> <apply><times/> <ci>a</ci> <ci>b</ci> <apply><sin/><ci>θ<!--GREEK SMALL LETTER THETA--></ci></apply> <ci type="vector"> N </ci> </apply> </apply> |
Sample Presentation
<mrow> <mrow><mi>A</mi><mo>×<!--MULTIPLICATION SIGN--></mo><mi>B</mi></mrow> <mo>=</mo> <mrow> <mi>a</mi> <mo>⁢<!--INVISIBLE TIMES--></mo> <mi>b</mi> <mo>⁢<!--INVISIBLE TIMES--></mo> <mrow><mi>sin</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>θ<!--GREEK SMALL LETTER THETA--></mi></mrow> <mo>⁢<!--INVISIBLE TIMES--></mo> <mi>N</mi> </mrow> </mrow> |
<scalarproduct/>
Class | binary-linalg |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | scalarproduct |
This element represents the scalar product function. It takes two vector arguments and returns a scalar value.
Content MathML
<apply><eq/> <apply><scalarproduct/> <ci type="vector">A</ci> <ci type="vector">B</ci> </apply> <apply><times/> <ci>a</ci> <ci>b</ci> <apply><cos/><ci>θ<!--GREEK SMALL LETTER THETA--></ci></apply> </apply> </apply> |
Sample Presentation
<mrow> <mrow><mi>A</mi><mo>.</mo><mi>B</mi></mrow> <mo>=</mo> <mrow> <mi>a</mi> <mo>⁢<!--INVISIBLE TIMES--></mo> <mi>b</mi> <mo>⁢<!--INVISIBLE TIMES--></mo> <mrow><mi>cos</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>θ<!--GREEK SMALL LETTER THETA--></mi></mrow> </mrow> </mrow> |
<outerproduct/>
Class | binary-linalg |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | outerproduct |
This element represents the outer product function. It takes two vector arguments and returns as value a matrix.
Content MathML
<apply><outerproduct/> <ci type="vector">A</ci> <ci type="vector">B</ci> </apply> |
Sample Presentation
<mrow><mi>A</mi><mo>⊗<!--CIRCLED TIMES--></mo><mi>B</mi></mrow> |
This section explains the use of the Constant and Symbol elements.
<integers/>
Class | constant-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | Z |
This element represents the set of integers, positive, negative and zero.
Content MathML
<apply><in/> <cn type="integer"> 42 </cn> <integers/> </apply> |
Sample Presentation
<mrow><mn>42</mn><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">Z</mi></mrow> |
<reals/>
Class | constant-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | R |
This element represents the set of real numbers.
Content MathML
<apply><in/> <cn type="real"> 44.997</cn> <reals/> </apply> |
Sample Presentation
<mrow> <mn>44.997</mn><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">R</mi> </mrow> |
<rationals/>
Class | constant-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | Q |
This element represents the set of rational numbers.
Content MathML
<apply><in/> <cn type="rational"> 22 <sep/>7</cn> <rationals/> </apply> |
Sample Presentation
<mrow> <mrow><mn>22</mn><mo>/</mo><mn>7</mn></mrow> <mo>∈<!--ELEMENT OF--></mo> <mi mathvariant="double-struck">Q</mi> </mrow> |
<naturalnumbers/>
Class | constant-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | N |
This element represents the set of natural numbers (including zero).
Content MathML
<apply><in/> <cn type="integer">1729</cn> <naturalnumbers/> </apply> |
Sample Presentation
<mrow> <mn>1729</mn><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">N</mi> </mrow> |
<complexes/>
Class | constant-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | C |
This element represents the set of complex numbers.
Content MathML
<apply><in/> <cn type="complex-cartesian">17<sep/>29</cn> <complexes/> </apply> |
Sample Presentation
<mrow> <mrow><mn>17</mn><mo>+</mo><mn>29</mn><mo>⁢<!--INVISIBLE TIMES--></mo><mi>i</mi></mrow> <mo>∈<!--ELEMENT OF--></mo> <mi mathvariant="double-struck">C</mi> </mrow> |
<primes/>
Class | constant-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | P |
This element represents the set of positive prime numbers.
Content MathML
<apply><in/> <cn type="integer">17</cn> <primes/> </apply> |
Sample Presentation
<mrow><mn>17</mn><mo>∈<!--ELEMENT OF--></mo><mi mathvariant="double-struck">P</mi></mrow> |
<exponentiale/>
Class | constant-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | e |
This element represents the base of the natural logarithm, approximately 2.718.
Content MathML
<apply><eq/> <apply><ln/><exponentiale/></apply> <cn>1</cn> </apply> |
Sample Presentation
<mrow> <mrow><mi>ln</mi><mo>⁡<!--FUNCTION APPLICATION--></mo><mi>e</mi></mrow> <mo>=</mo> <mn>1</mn> </mrow> |
<imaginaryi/>
Class | constant-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | i |
This element represents the mathematical constant which is the square root of -1, commonly written i
Content MathML
<apply><eq/> <apply><power/><imaginaryi/><cn>2</cn></apply> <cn>-1</cn> </apply> |
Sample Presentation
<mrow><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mn>-1</mn></mrow> |
<notanumber/>
Class | constant-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | NaN |
This element represents the notion of not-a-number, i.e. the result of an ill-posed floating computation. See [IEEE754].
Content MathML
<apply><eq/> <apply><divide/><cn>0</cn><cn>0</cn></apply> <notanumber/> </apply> |
Sample Presentation
<mrow> <mrow><mn>0</mn><mo>/</mo><mn>0</mn></mrow> <mo>=</mo> <mi>NaN</mi> </mrow> |
<true/>
Class | constant-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | true |
This element represents the Boolean value true, i.e. the logical constant for truth.
Content MathML
<apply><eq/> <apply><or/> <true/> <ci type="boolean">P</ci> </apply> <true/> </apply> |
Sample Presentation
<mrow> <mrow><mi>true</mi><mo>∨<!--LOGICAL OR--></mo><mi>P</mi></mrow> <mo>=</mo> <mi>true</mi> </mrow> |
<false/>
Class | constant-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | false |
This element represents the Boolean value false, i.e. the logical constant for falsehood.
Content MathML
<apply><eq/> <apply><and/> <false/> <ci type="boolean">P</ci> </apply> <false/> </apply> |
Sample Presentation
<mrow> <mrow><mi>false</mi><mo>∧<!--LOGICAL AND--></mo><mi>P</mi></mrow> <mo>=</mo> <mi>false</mi> </mrow> |
<emptyset/>
Class | constant-set |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | emptyset, emptyset |
This element is used to represent the empty set, that is the set which contains no members.
Content MathML
<apply><neq/> <integers/> <emptyset/> </apply> |
Sample Presentation
<mrow> <mi mathvariant="double-struck">Z</mi><mo>≠<!--NOT EQUAL TO--></mo><mi>∅<!--EMPTY SET--></mi> </mrow> |
Mapping to Strict Markup
In some situations, it may be clear from context that emptyset
corresponds to the emptyset
However, as there is no method other than annotation for an author to explicitly indicate this,
it is always acceptable to translate to the emptyset symbol from the set1 CD.
<pi/>
Class | constant-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | pi |
This element represents pi, approximately 3.142, which is the ratio of the circumference of a circle to its diameter.
Content MathML
<apply><approx/> <pi/> <cn type="rational">22<sep/>7</cn> </apply> |
Sample Presentation
<mrow> <mi>π<!--GREEK SMALL LETTER PI--></mi> <mo>≃<!--ASYMPTOTICALLY EQUAL TO--></mo> <mrow><mn>22</mn><mo>/</mo><mn>7</mn></mrow> </mrow> |
<eulergamma/>
Class | constant-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | gamma |
This element denotes the gamma constant, approximately 0.5772.
Content MathML
<apply><approx/> <eulergamma/> <cn>0.5772156649</cn> </apply> |
Sample Presentation
<mrow><mi>γ<!--GREEK SMALL LETTER GAMMA--></mi><mo>≃<!--ASYMPTOTICALLY EQUAL TO--></mo><mn>0.5772156649</mn></mrow> |
<infinity/>
Class | constant-arith |
---|---|
Attributes | CommonAtt, DefEncAtt |
Content | Empty |
OM Symbols | infinity |
This element represents the notion of infinity.
Content MathML
<infinity/> |
Sample Presentation
<mi>∞<!--INFINITY--></mi> |
<declare>
Attributes | CommonAtt, type, scope, occurrence, definitionURL, encoding |
---|---|
type Attribute
|
defines the MathML element type of the identifier declared. |
scope Attribute
|
defines the scope of application of the declaration. |
nargs Attribute
|
number of arguments for function declarations. |
occurrence Attribute values
|
"prefix" | "infix" | "function-model" |
definitionURL Attribute
|
URI pointing to detailed semantics of the function. |
encoding Attribute
|
syntax of the detailed semantics of the function. |
Content | ContExp, ContExp? |
MathML2 provided the declare
element to bind properties like
types to symbols and variables and to define abbreviations for structure sharing. This
element is deprecated in MathML 3. Structure sharing can obtained via the share
element (see Section 4.2.7 Structure Sharing <share>
for details).
<reln>
Content | ContExp* |
---|
MathML1 provided the reln
element to construct an equation or relation.
This usage was deprecated in MathML 2.0 in favor of the more generally usable apply
.
<fn>
Content | ContExp |
---|
MathML1 provided the fn
element to extend the collection of known mathematical
functions. This usage was deprecated in MathML 2.0 in favor of the more generally
applicable csymbol
.
MathML 3 assigns semantics to content markup by defining a mapping to Strict Content MathML. Strict MathML, in turn, is in one-to-one correspondence with OpenMath, and the subset of OpenMath expressions obtained from content MathML expressions in this fashion all have well-defined semantics via the standard OpenMath Content Dictionary set. Consequently, the mapping of arbitrary content MathML expressions to equivalent Strict Content MathML plays a key role in underpinning the meaning of content MathML.
The mapping of arbitrary content MathML into Strict content MathML is defined algorithmically. The algorithm is described below as a collection of rewrite rules applying to specific non-Strict constructions. The individual rewrite transformations have been described in detail in context above. The goal of this section is to outline the complete algorithm in one place.
The algorithm is a sequence of nine steps. Each step is applied repeatedly to rewrite the input until no further application is possible. Note that in many programming languages, such as XSLT, the natural implementation is as a recursive algorithm, rather than the multi-pass implementation suggested by the description below. The translation to XSL is straightforward and produces the same eventual Strict Content MathML. However, because the overall structure of the multi-pass algorithm is clearer, that is the formulation given here.
To transform an arbitrary content MathML expression into Strict Content MathML, apply each of the following rules in turn to the input expression until all instances of the target constructs have been eliminated:
Rewrite non-strict bind
and elminate deprecated elements:
Change the outer bind
tags
in binding expressions to apply
if they have qualifiers or multiple
children. This simplifies the algorithm by allowing the subsequent rules to be applied
to non-strict binding expressions without case distinction. Note
that the later
rules will change the apply
elements introduced in this step back to
bind
elements. Also in this step, deprecated reln
elements are rewritten to apply
, and fn
elements are replaced by
the child expressions they enclose.
Apply special case rules for idiomatic uses of qualifiers:
Rewrite derivatives with rules Rewrite: diff, Rewrite: nthdiff, and Rewrite: partialdiffdegree to explicate the binding status of the variables involved.
Rewrite integrals with the rules Rewrite: int, Rewrite: defint
and Rewrite: defint limits to disambiguate the status
of bound and free variables and of the orientation of the range of integration if
it is given as a lowlimit
/uplimit
pair.
Rewrite limits as described in Rewrite: tendsto and Rewrite: limits condition.
Rewrite sums and products as described in
Section 4.4.6.1 Sum <sum/>
and Section 4.4.6.2 Product <product/>
.
Rewrite roots as described in Section 4.4.2.11 Root <root/>
.
Rewrite logarithms as described in Section 4.4.7.7 Logarithm <log/>
, <logbase>
.
Rewrite moments as described in Section 4.4.8.6 Moment <moment/>
, <momentabout>
.
Rewrite Qualifiers to domainofapplication
:
These rules rewrite all apply
constructions using bvar
and
qualifiers to those using only the general domainofapplication
qualifier.
Intervals: Rewrite qualifiers given as interval
and
lowlimit
/uplimit
to intervals of integers via
Rewrite: interval qualifier.
Multiple condition
s: Rewrite multiple condition
qualifiers to a single one by taking their conjunction. The resulting compound
condition
is then rewritten to domainofapplication
according
to rule Rewrite: condition.
Multiple domainofapplication
s: Rewrite multiple
domainofapplication
qualifiers to a single one by taking the
intersection of the specified domains.
Normalize Container Markup:
Rewrite sets and lists by the rule Rewrite: n-ary setlist domainofapplication.
Rewrite interval, vectors, matrices, and matrix rows
as described in Section 4.4.1.1 Interval <interval>
, Section 4.4.9.1 Vector <vector>
,
Section 4.4.9.2 Matrix <matrix>
and Section 4.4.9.3 Matrix row <matrixrow>
. Note any qualifiers will have been rewritten to domainofapplication
and will be further rewritten in Step 6.
Rewrite lambda expressions by the rules Rewrite: lambda and Rewrite: lambda domainofapplication
Rewrite piecewise functions as described in Section 4.4.1.9 Piecewise declaration <piecewise>
, <piece>
, <otherwise>
.
Apply Special Case Rules for Operators using domainofapplication
Qualifiers:
This step deals with the special cases for the operators introduced in
Section 4.4 Content MathML for Specific Operators and Constants. There are different classes of special cases to be taken into account:
Rewrite min
, max
, mean
and similar n-ary/unary operators
by the rules Rewrite: n-ary unary set, Rewrite: n-ary unary domainofapplication
and Rewrite: n-ary unary single.
Rewrite the quantifiers forall
and exists
used with domainofapplication
to expressions using implication and conjunction by the rule Rewrite: quantifier.
Rewrite integrals used with a domainofapplication
element (with or without a bvar
)
according to the rules Rewrite: int and
Rewrite: defint.
Rewrite sums and products used with a domainofapplication
element
(with or without a bvar
) as described in
Section 4.4.6.1 Sum <sum/>
and Section 4.4.6.2 Product <product/>
.
Eliminate domainofapplication
: At this stage, any
apply
has at most one domainofapplication
child and special cases have been addressed. As
domainofapplication
is not Strict Content MathML, it is rewritten
into an application of a restricted function via the rule
Rewrite: restriction if the apply
does not contain
a bvar
child.
into an application of the predicate_on_list symbol via the rules Rewrite: n-ary relations and Rewrite: n-ary relations bvar if used with a relation.
into a construction with the apply_to_list symbol via the general rule Rewrite: n-ary domainofapplication for general n-ary operators.
into a construction using the suchthat symbol
from the set1 content dictionary in an apply
with bound
variables via the Rewrite: apply bvar domainofapplication rule.
Rewrite non-strict token elements:
Rewrite numbers represented as cn
elements where the type
attribute is one of "e-notation", "rational",
"complex-cartesian", "complex-polar",
"constant" as strict cn
via rules
Rewrite: cn sep, Rewrite: cn based_integer
and Rewrite: cn constant.
Rewrite any ci
, csymbol
or cn
containing
presentation MathML to semantics
elements with rules
Rewrite: cn presentation mathml and Rewrite: ci presentation mathml and
the analogous rule for csymbol
.
Rewrite operators: Rewrite any remaining operator defined in Section 4.4 Content MathML for Specific Operators and Constants
to a csymbol
referencing the symbol identified in the syntax table by the rule
Rewrite: element. As noted in the descriptions of each
operator element, some require special case rules to determine the proper choice of symbol.
Some cases of particular note are:
The order of the arguments for the
selector
operator must be
rewritten, and the symbol depends on the type of the arguments.
The choice of symbol for the minus
operator depends on the number of the arguments.
The choice of symbol for some set operators depends on the values of
the type
of the arguments.
The choice of symbol for some statistical operators depends on the values of the types of the arguments.
Rewrite non-strict attributes:
Rewrite the type
attribute:
At this point, all elements
that accept the type
, other than ci
and csymbol
, should have been
rewritten into Strict Content Markup equivalents without type
attributes,
where type information is reflected in the choice of operator symbol. Now rewrite remaining
ci
and csymbol
elements with a type
attribute to a
strict expression with semantics
according to rules
Rewrite: ci type annotation and Rewrite: csymbol type annotation.
Rewrite definitionURL
and encoding
attributes:
If the definitionURL
and encoding
attributes on a
csymbol
element can be interpreted as a reference to a
content dictionary (see Section 4.2.3.2 Non-Strict uses of <csymbol>
for details), then
rewrite to reference the content dictionary by the cd
attribute instead.
Rewrite attributes: Rewrite any element with attributes that are
not allowed in strict markup to a semantics
construction with
the element without these attributes as the first child and the attributes in
annotation
elements by rule Rewrite: attributes.
MathML markup can be combined with other markup languages, and
these mixing constructions are realized by the semantic annotation
elements. The semantic annotation elements provide an important tool
for making associations between alternate representations of an
expression, and for associating semantic properties and other
attributions with a MathML expression. These elements allow
presentation markup and content markup to be combined in several
different ways. One method, known as mixed markup, is to
intersperse content and presentation elements in what is essentially a
single tree. Another method, known as parallel markup,
is to provide both explicit presentation markup and content markup in
a pair of markup expressions, combined by a single semantics
element.
An important concern of MathML is to represent associations between presentation and content markup forms for an expression. Representing associations between MathML expressions and data of other kinds is also important in many contexts. For this reason, MathML provides a general framework for annotation. A MathML expression may be decorated with a sequence of pairs made up of a symbol that indicates the kind of annotation, known as the annotation key, and associated data, known as the annotation value.
The semantics
, annotation
, and
annotation-xml
elements are used together to represent
annotations in MathML. The semantics
element provides the
container for a expression and its annotations. The
annotation
element is the container for text
annotations, and the annotation-xml
element is used for
structured
annotations. The semantics
element contains the expression
being annotated as its first child, followed by a sequence of zero or
more annotation
and/or annotation-xml
elements.
<semantics> <mrow> <mrow> <mi>sin</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>x</mi></mfenced> </mrow> <mo>+</mo> <mn>5</mn> </mrow> <annotation encoding="application/x-tex"> \sin x + 5 </annotation> <annotation-xml encoding="application/openmath+xml"> <OMA xmlns="http://www.openmath.org/OpenMath"> <OMS cd="arith1" name="plus"/> <OMA><OMS cd="transc1" name="sin"/><OMV name="x"/></OMA> <OMI>5</OMI> </OMA> </annotation-xml> </semantics> |
Note that this example makes use of the namespace extensibility
that is only available in the XML syntax of MathML. If this example is included in an HTML document
then it would be considered invalid and the OpenMath elements would be parsed as
elements un the MathML namespace. See Section 5.2.3.3 Using annotation-xml
in HTML documents for details.
The semantics
element is considered to be both a presentation
element and a content element, and may be used in either context.
All MathML processors should process the semantics
element,
even if they only process one of these two subsets of MathML.
An annotation key specifies the relationship between an expression and an annotation. Many kinds of relationships are possible. Examples include alternate representations, specification or clarification of semantics, type information, rendering hints, and private data intended for specific processors. The annotation key is the primary means by which a processor determines whether or not to process an annotation.
The logical relationship between an expression and an annotation can have a significant impact on the proper processing of the expression. For example, a particular annotation form, called semantic attributions, cannot be ignored without altering the meaning of the annotated expression, at least in some processing contexts. On the other hand, alternate representations do not alter the meaning of an expression, but may alter the presentation of the expression as they are frequently used to provide rendering hints. Still other annotations carry private data or metadata that are useful in a specific context, but do not alter either the semantics or the presentation of the expression.
In MathML 3, annotation keys are defined as symbols in Content Dictionaries, and are specified using
of the cd
and name
attributes on the
annotation
and annotation-xml
elements. For backward
compatibility with MathML 2, an annotation key may also be referenced
using the definitionURL
attribute as an alternative to the
cd
and name
attributes. Further details on
referencing symbols in Content Dictionaries are discussed in Section 4.2.3 Content Symbols <csymbol>
. The symbol definition in a Content Dictionary
for an annotation key may have a role
property. Two
particular roles are relevant for annotations: a role of
"attribution" identifies a generic annotation that can
be ignored without altering the meaning of the annotated term, and a
role of "semantic-attribution" indicates that the annotation
is a semantic annotation, that is, the annotation cannot be ignored
without potentially altering the meaning of the expression.
MathML 3 provides two predefined annotation keys for the most common kinds of annotations: alternate-representation and contentequiv defined in the mathmlkeys content dictionary. The alternate-representation annotation key specifies that the annotation value provides an alternate representation for an expression in some other markup language, and the contentequiv annotation key specifies that the annotation value provides a semantically equivalent alternative for the annotated expression. Further details about the use of these keys is given in the sections below.
The default annotation key is alternate-representation when no annotation key is
explicitly specified on an annotation
or
annotation-xml
element.
Typically, annotation keys specify only the logical nature of the
relationship between an expression and an annotation. The data format
for an annotation is indicated with the encoding
attribute. In MathML 2, the encoding
attribute was the
primary information that a processor could use to determine whether or
not it could understand an annotation. For backward compatibility,
processors are encouraged to examine both the annotation key and
encoding
attribute. In
particular, MathML 2 specified the predefined encoding values
MathML
, MathML-Content
, and
MathML-Presentation
. The MathML
encoding
value is used to indicate an annotation-xml
element contains
a MathML expression. The use of the other values is more specific, as
discussed in following sections.
While the predefined alternate-representation and contentequiv keys cover many common use cases, user communities are encouraged to define and standardize additional content dictionaries as necessary. Annotation keys in user-defined, public Content Dictionaries are preferred over private encoding attribute value conventions, since content dictionaries are more expressive, more open and more maintainable than private encoding values. However, for backward compatibility with MathML 2, the encoding attribute may also be used.
Alternate representation annotations are most often used to provide renderings for an expression, or to provide an equivalent representation in another markup language. In general, alternate representation annotations do not alter the meaning of the annotated expression, but may alter its presentation.
A particularly important case is the use of a presentation MathML
expression to indicate a preferred rendering for a content MathML
expression. This case may be represented by labeling the annotation
with the application/mathml-presentation+xml
value for
the encoding
attribute. For backward compatibility with
MathML 2.0, this case can also be represented with the equivalent
MathML-Presentation
value for the encoding
attribute. Note that when a presentation MathML annotation is
present in a semantics
element, it may be used as the
default rendering of the semantics
element, instead of
the default rendering of the first child.
In the example below, the semantics
element binds together
various alternate representations for a content MathML expression.
The presentation MathML annotation may be used as the
default rendering, while the other annotations give representations
in other markup languages. Since no attribution keys are explicitly
specified, the default annotation key
alternate-representation applies
to each of the annotations.
<semantics> <apply> <plus/> <apply><sin/><ci>x</ci></apply> <cn>5</cn> </apply> <annotation-xml encoding="MathML-Presentation"> <mrow> <mrow> <mi>sin</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced open="(" close=")"><mi>x</mi></mfenced> </mrow> <mo>+</mo> <mn>5</mn> </mrow> </annotation-xml> <annotation encoding="application/x-maple"> sin(x) + 5 </annotation> <annotation encoding="application/vnd.wolfram.mathematica"> Sin[x] + 5 </annotation> <annotation encoding="application/x-tex"> \sin x + 5 </annotation> <annotation-xml encoding="application/openmath+xml"> <OMA xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS cd="arith1" name="plus"/> <OMA><OMS cd="transc1" name="sin"/><OMV name="x"/></OMA> <OMI>5</OMI> </OMA> </OMA> </annotation-xml> </semantics> |
Note that this example makes use of the namespace extensibility
that is only available in the XML syntax of MathML. If this example is included in an HTML document
then it would be considered invalid and the OpenMath elements would be parsed as
elements un the MathML namespace. See Section 5.2.3.3 Using annotation-xml
in HTML documents for details.
Content equivalent annotations provide additional computational information about an expression. Annotations with the contentequiv key cannot be ignored without potentially changing the behavior of an expression.
An important case arises when a content MathML annotation is used
to disambiguate the meaning of a presentation MathML expression.
This case may be represented by labeling the annotation with the
application/mathml-content+xml
value for the
encoding
attribute. In
MathML 2, this type of annotation was represented with the equivalent
MathML-Content
value for the encoding
attribute,
so processors are urged to support this usage for backward compatibility.
A content MathML annotation, whether in MathML 2 or 3, may be used for
other purposes as well, such as for other kinds of semantic assertions.
Consequently, in MathML 3, the
contentequiv annotation key should
be used to make an explicit assertion that the annotation provides a
definitive content markup equivalent for an expression.
In the example below, an ambiguous presentation MathML expression
is annotated with a MathML-Content
annotation clarifying
its precise meaning.
<semantics> <mrow> <mrow> <mi>a</mi> <mfenced open="(" close=")"> <mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow> </mfenced> </mrow> </mrow> <annotation-xml cd="mathmlkeys" name="contentequiv" encoding="MathML-Content"> <apply> <ci>a</ci> <apply><plus/><ci>x</ci><cn>5</cn></apply> </apply> </annotation-xml> </semantics> |
In the usual case, each annotation element includes either character data
content (in the case of annotation
) or XML markup data (in the case
of annotation-xml
) that represents the annotation value.
There is no restriction on the type of annotation that may appear within a
semantics
element. For example, an annotation could provide a
TEX encoding, a linear input form for a computer algebra system,
a rendered image, or detailed mathematical type information.
In some cases the alternative children of a semantics
element
are not an essential part of the behavior of the annotated expression, but
may be useful to specialized processors. To enable the availability of
several annotation formats in a more efficient manner, a semantics
element may contain empty annotation
and annotation-xml
elements that provide encoding
and src
attributes
to specify an external location for the annotation value associated with
the annotation. This type of annotation is known as an annotation
reference.
<semantics> <mfrac><mi>a</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfrac> <annotation encoding="image/png" src="333/formula56.png"/> <annotation encoding="application/x-maple" src="333/formula56.ms"/> </semantics>
Processing agents that anticipate that consumers of exported markup may not be able to retrieve the external entity referenced by such annotations should request the content of the external entity at the indicated location and replace the annotation with its expanded form.
An annotation reference follows the same rules as for other annotations to determine the annotation key that specifies the relationship between the annotated object and the annotation value.
This section explains the semantic mapping elements semantics
,
annotation
, and annotation-xml
. These elements associate
alternate representations for a presentation or content expression, or
associate semantic or other attributions that may modify the meaning of
the annotated expression.
<semantics>
element
The semantics
element is the container element that
associates annotations with a MathML expression. The
semantics
element has as its first child the expression to be
annotated. Any MathML expression may appear as the first child of the
semantics
element. Subsequent annotation
and
annotation-xml
children enclose the annotations.
An annotation represented in XML is enclosed in an
annotation-xml
element. An annotation represented
in character data is enclosed in an annotation
element.
As noted above, the semantics
element is considered to be
both a presentation element and a content element, since it can act
as either, depending on its content. Consequently, all MathML
processors should process the semantics
element, even if they
process only presentation markup or only content markup.
The default rendering of a semantics
element is the default
rendering of its first child. A renderer may use the information contained
in the annotations to customize its rendering of the annotated element.
<semantics> <mrow> <mrow> <mi>sin</mi> <mo>⁡<!--FUNCTION APPLICATION--></mo> <mfenced><mi>x</mi></mfenced> </mrow> <mo>+</mo> <mn>5</mn> </mrow> <annotation-xml cd="mathmlkeys" name="contentequiv" encoding="MathML-Content"> <apply> <plus/> <apply><sin/><ci>x</ci></apply> <cn>5</cn> </apply> </annotation-xml> <annotation encoding="application/x-tex"> \sin x + 5 </annotation> </semantics> |
Name | values | default |
---|---|---|
definitionURL | URI | none |
The location of an external source for semantic information | ||
encoding | string | none |
The encoding of the external semantic information |
The semantics
element takes the definitionURL
and
encoding
attributes, which reference an external source for some
or all of the semantic information for the annotated element, as modified
by the annotation. The use of these attributes on the semantics
element is deprecated in MathML3.
<annotation>
element
The annotation
element is the container element for a semantic
annotation whose representation is parsed character data in a non-XML
format. The annotation
element should contain the character
data for the annotation, and should not contain XML markup elements.
If the annotation contains one of the XML reserved characters
&
, <
then these characters must
be encoded using an entity reference or
(in the XML syntax)
an XML CDATA
section.
Name | values | default |
---|---|---|
definitionURL | URI | none |
The location of the annotation key symbol | ||
encoding | string | none |
The encoding of the semantic information in the annotation | ||
cd | string | mathmlkeys |
The content dictionary that contains the annotation key symbol | ||
name | string | alternate-representation |
The name of the annotation key symbol | ||
src | URI | none |
The location of an external source for semantic information |
Taken together, the cd
and name
attributes
specify the annotation key symbol, which identifies the relationship
between the annotated element and the annotation, as described in
Section 5.1.1 Annotation elements. The definitionURL
attribute provides an alternate way to reference the annotation key
symbol as a single attribute. If none of these attributes are present,
the annotation key symbol is the symbol
alternate-representation
from the mathmlkeys content dictionary.
The encoding
attribute describes the content type of the
annotation. The value of the encoding
attribute may contain
a media type that identifies the data format for the encoding data. For
data formats that do not have an associated media type, implementors may
choose a self-describing character string to identify their content type.
The src
attribute provides a mechanism to attach external
entities as annotations on MathML expressions.
<annotation encoding="image/png" src="333/formula56.png"/>
The annotation
element is a semantic mapping element that may
only be used as a child of the semantics
element. While there is
no default rendering for the annotation
element, a renderer may
use the information contained in an annotation to customize its rendering
of the annotated element.
<annotation-xml>
element
The annotation-xml
element is the container element for a
semantic annotation whose representation is structured markup. The annotation-xml
element should contain the markup
elements, attributes, and character data for the annotation.
Name | values | default |
---|---|---|
definitionURL | URI | none |
The location of the annotation key symbol | ||
encoding | string | none |
The encoding of the semantic information in the annotation | ||
cd | string | mathmlkeys |
The content dictionary that contains the annotation key symbol | ||
name | string | alternate-representation |
The name of the annotation key symbol | ||
src | URI | none |
The location of an external source for semantic information |
Taken together, the cd
and name
attributes
specify the annotation key symbol, which identifies the relationship
between the annotated element and the annotation, as described in
Section 5.1.1 Annotation elements. The definitionURL
attribute provides an alternate way to reference the annotation key
symbol as a single attribute. If none of these attributes are present,
the annotation key symbol is the symbol
alternate-representation
from the mathmlkeys content dictionary.
The encoding
attribute describes the content type of the
annotation. The value of the encoding
attribute may contain
a media type that identifies the data format for the encoding data. For
data formats that do not have an associated media type, implementors may
choose a self-describing character string to identify their content type.
In particular, as described above and in Section 6.2.4 Names of MathML Encodings, MathML specifies
MathML
, MathML-Presentation
, and
MathML-Content
as predefined values for the
encoding
attribute. Finally, The src
attribute provides a mechanism to attach external XML entities as
annotations on MathML expressions.
<annotation-xml cd="mathmlkeys" name="contentequiv" encoding="MathML-Content"> <apply> <plus/> <apply><sin/><ci>x</ci></apply> <cn>5</cn> </apply> </annotation-xml> <annotation-xml encoding="application/openmath+xml"> <OMA xmlns="http://www.openmath.org/OpenMath"> <OMS cd="arith1" name="plus"/> <OMA><OMS cd="transc1" name="sin"/><OMV name="x"/></OMA> <OMI>5</OMI> </OMA> </annotation-xml>
When the MathML is being parsed as XML and the annotation value is represented in an XML dialect other than MathML, the namespace for the XML markup for the annotation should be identified by means of namespace attributes and/or namespace prefixes on the annotation value. For instance:
<annotation-xml encoding="application/xhtml+xml"> <html xmlns="http://www.w3.org/1999/xhtml"> <head><title>E</title></head> <body> <p>The base of the natural logarithms, approximately 2.71828.</p> </body> </html> </annotation-xml>
The annotation-xml
element is a semantic mapping element that may only be used
as a child of the semantics
element. While there is no default rendering for the
annotation-xml
element, a renderer may use the information contained in an
annotation to customize its rendering of the annotated element.
annotation-xml
in HTML documents
Note that the Namespace extensibility used in the above examples
may not be available if the MathML is not being treated as an XML document. In particular HTML parsers
treat xmlns
attributes as ordinary attributes, so the OpenMath example would be classified as invalid
by an HTML validator. The OpenMath elements would still be parsed as children of the annotation-xml
element, however they would be placed in the MathML namespace. The above examples are not rendered in the HTML version of this specification, to ensure that that document is
a valid HTML5 document.
The details of the HTML parser handling of annotation-xml
is specified in [HTML5] and summarized in Section 6.4.3 Mixing MathML and HTML, however the main differences from the behavior of an XML parser that affect MathML annotations are that the HTML parser
does not treat xmlns
attributes, nor :
in element names as special and has built-in rules determining whether the three "known" namespaces, HTML, SVG or MathML
are used.
If the annotation-xml
has an encoding
attribute that is (ignoring case differences) "text/html" or "annotation/xhtml+xml" then the content is parsed as HTML and
placed (initially) in the HTML namespace.
Otherwise it is parsed as foreign content and parsed in a more XML-like manner (like MathML itself in HTML) in which />
signifies an empty element. Content will be placed in the MathML namespace.
If any recognised HTML element appears in this foreign content annotation the HTML parser will effectively termnate the math
expression, closing all open elements until the math
element is closed, and then process the nested HTML as if it were not inside the math context. Any following MathML elements
will then not render correctly as they are not in a math context, or in the MathML namespace.
These issues mean that the following example is valid whether parsed by an XML or HTML parser:
<math> <semantics> <mi>a</mi> <annotation-xml encoding="text/html"> <span>xxx</span> </annotation-xml> </semantics> <mo>+</mo> <mi>b</mi> </math>
However the if the encoding
attribute is omitted then the expression
is only valid if parsed as XML:
<math> <semantics> <mi>a</mi> <annotation-xml> <span>xxx</span> </annotation-xml> </semantics> <mo>+</mo> <mi>b</mi> </math>
If the above is parsed by an HTML parser it produces a result equivalent to the following invalid input, where the span
element has caused all MathML elements to be prematurely closed. The remaining MathML elements following the span
are no longer contained within <math>
so will be parsed as unknown HTML elements and render incorrectly.
<math xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mi>a</mi> <annotation-xml> </annotation-xml> </semantics> </math> <span xmlns="http://www.w3.org/1999/xhtml">xxx</span> <mo xmlns="http://www.w3.org/1999/xhtml">+</mo> <mi xmlns="http://www.w3.org/1999/xhtml">b</mi>
Note here that the HTML span
element has
caused all open MathML elements to be prematurely closed, resulting in
the following MathML elements being treated as unknown HTML elements
as they are no longer descendents of math
. See Section 6.4.3 Mixing MathML and HTML for more details of the parsing of MathML in HTML.
Any use of elements in other vocabularies (such as the OpenMath
examples above) is considered invalid in HTML. If
validity is not a strict requirement it is possible to use such
elements but they will be parsed as elements on the
MathML namespace. Documents SHOULD NOT use namespace
prefixes and element names containing colon (:
) as the
element nodes produced by the HTML parser with have local names
containing a colon, which can not be constructed by a namespace aware
XML parser. Rather than use such foreign annotations, when using an HTML parser
it is better to encode the annotation using the existing
vocabulary. For example as shown in Chapter 4 Content Markup OpenMath
may be encoded faithfuly as Strict Content
MathML. Similarly RDF annotations could be encoded using RDFa
in text/html
annotation or (say) N3 notation in
annotation
rather than using RDF/XML encoding in an
annotation-xml
element.
Presentation markup encodes the notational structure of an expression. Content markup encodes the functional structure of an expression. In certain cases, a particular application of MathML may require a combination of both presentation and content markup. This section describes specific constraints that govern the use of presentation markup within content markup, and vice versa.
Presentation markup may be embedded within content markup so long as the resulting expression retains an unambiguous function application structure. Specifically, presentation markup may only appear in content markup in three ways:
within ci
and cn
token elements
within the csymbol
element
within the semantics
element
Any other presentation markup occurring within content markup is a MathML error. More detailed discussion of these three cases follows:
ci
and cn
are permitted to
contain any sequence of MathML characters (defined in Chapter 7 Characters, Entities and Fonts)
and/or presentation elements. Contiguous blocks of MathML characters in
ci
or cn
elements are treated as if wrapped in
mi
or mn
elements, as appropriate, and the resulting
collection of presentation elements is rendered as if wrapped in an
implicit mrow
element.
csymbol
element.
csymbol
element may contain either MathML characters
interspersed with presentation markup, or content markup. It is a MathML
error for a csymbol
element to contain both presentation and
content elements. When the csymbol
element contains
character data and presentation markup, the same rendering rules that apply
to the token elements ci
and cn
should be used.
semantics
element.
semantics
element is to
provide a mechanism for incorporating arbitrary MathML expressions into
content markup in a semantically meaningful way. In particular, any valid
presentation expression can be embedded in a content expression by placing
it as the first child of a semantics
element. The meaning of this
wrapped expression should be indicated by one or more annotation elements
also contained in the semantics
element.
Content markup may be embedded within presentation markup so long as the resulting expression has an unambiguous rendering. That is, it must be possible, in principle, to produce a presentation markup fragment for each content markup fragment that appears in the combined expression. The replacement of each content markup fragment by its corresponding presentation markup should produce a well-formed presentation markup expression. A presentation engine should then be able to process this presentation expression without reference to the content markup bits included in the original expression.
In general, this constraint means that each embedded content expression
must be well-formed, as a content expression, and must be able to stand
alone outside the context of any containing content markup element. As
a result, the following content elements may not appear as an immediate
child of a presentation element:
annotation
, annotation-xml
,
bvar
, condition
, degree
,
logbase
, lowlimit
, uplimit
.
In addition, within presentation markup, content markup may not appear within presentation token elements.
Some applications are able to use both presentation
and content information. Parallel markup is a way to
combine two or more markup trees for the same mathematical expression.
Parallel markup is achieved with the semantics
element.
Parallel markup for an expression may appear on its own, or as part
of a larger content or presentation tree.
In many cases, the goal is to provide presentation markup and content
markup for a mathematical expression as a whole.
A single semantics
element may be used to pair two markup trees,
where one child element provides the presentation markup, and the
other child element provides the content markup.
The following example encodes the Boolean arithmetic expression (a+b)(c+d) in this way.
<semantics> <mrow> <mrow><mo>(</mo><mi>a</mi> <mo>+</mo> <mi>b</mi><mo>)</mo></mrow> <mo>⁢<!--INVISIBLE TIMES--></mo> <mrow><mo>(</mo><mi>c</mi> <mo>+</mo> <mi>d</mi><mo>)</mo></mrow> </mrow> <annotation-xml encoding="MathML-Content"> <apply><and/> <apply><xor/><ci>a</ci> <ci>b</ci></apply> <apply><xor/><ci>c</ci> <ci>d</ci></apply> </apply> </annotation-xml> </semantics> |
Note that the above markup annotates the presentation markup as
the first child element, with the content markup as part of the
annotation-xml
element. An equivalent form could be given
that annotates the content markup as the first child element, with
the presentation markup as part of the annotation-xml
element.
To accommodate applications that must process sub-expressions of large
objects, MathML supports cross-references between the branches of a
semantics
element to identify corresponding sub-structures.
These cross-references are established by the use of the id
and xref
attributes within a semantics
element.
This application of the id
and xref
attributes within
a semantics
element should be viewed as best practice to enable
a recipient to select arbitrary sub-expressions in each alternative
branch of a semantics
element. The id
and
xref
attributes may be placed on MathML elements of any type.
The following example demonstrates cross-references for the Boolean arithmetic expression (a+b)(c+d).
<semantics> <mrow id="E"> <mrow id="E.1"> <mo id="E.1.1">(</mo> <mi id="E.1.2">a</mi> <mo id="E.1.3">+</mo> <mi id="E.1.4">b</mi> <mo id="E.1.5">)</mo> </mrow> <mo id="E.2">⁢<!--INVISIBLE TIMES--></mo> <mrow id="E.3"> <mo id="E.3.1">(</mo> <mi id="E.3.2">c</mi> <mo id="E.3.3">+</mo> <mi id="E.3.4">d</mi> <mo id="E.3.5">)</mo> </mrow> </mrow> <annotation-xml encoding="MathML-Content"> <apply xref="E"> <and xref="E.2"/> <apply xref="E.1"> <xor xref="E.1.3"/><ci xref="E.1.2">a</ci><ci xref="E.1.4">b</ci> </apply> <apply xref="E.3"> <xor xref="E.3.3"/><ci xref="E.3.2">c</ci><ci xref="E.3.4">d</ci> </apply> </apply> </annotation-xml> </semantics> |
An id
attribute and associated xref
attributes
that appear within the same semantics
element establish the
cross-references between corresponding sub-expressions.
For parallel markup, all of the id
attributes referenced by any xref
attribute should be in the same branch of an enclosing
semantics
element. This constraint guarantees that the
cross-references do not create unintentional cycles. This restriction
does not exclude the use of id
attributes within
other branches of the enclosing semantics
element. It does,
however, exclude references to these other id
attributes
originating from the same semantics
element.
There is no restriction on which branch of the semantics
element
may contain the destination id
attributes. It is up to the
application to determine which branch to use.
In general, there will not be a one-to-one correspondence between nodes
in parallel branches. For example, a presentation tree may contain elements,
such as parentheses, that have no correspondents in the content tree. It is
therefore often useful to put the id
attributes on the branch with
the finest-grained node structure. Then all of the other branches will have
xref
attributes to some subset of the id
attributes.
In absence of other criteria, the first branch of the semantics
element is a sensible choice to contain the id
attributes.
Applications that add or remove annotations will then not have to re-assign
these attributes as the annotations change.
In general, the use of id
and xref
attributes allows
a full correspondence between sub-expressions to be given in text that is
at most a constant factor larger than the original. The direction of the
references should not be taken to imply that sub-expression selection is
intended to be permitted only on one child of the semantics
element.
It is equally feasible to select a subtree in any branch and
to recover the corresponding subtrees of the other branches.
Parallel markup with cross-references may be used in any XML-encoded branch of the semantic annotations, as shown by the following example where the Boolean expression of the previous section is annotated with OpenMath markup that includes cross-references:
<semantics> <mrow id="EE"> <mrow id="EE.1"> <mo id="EE.1.1">(</mo> <mi id="EE.1.2">a</mi> <mo id="EE.1.3">+</mo> <mi id="EE.1.4">b</mi> <mo id="EE.1.5">)</mo> </mrow> <mo id="EE.2">⁢<!--INVISIBLE TIMES--></mo> <mrow id="EE.3"> <mo id="EE.3.1">(</mo> <mi id="EE.3.2">c</mi> <mo id="EE.3.3">+</mo> <mi id="EE.3.4">d</mi> <mo id="EE.3.5">)</mo> </mrow> </mrow> <annotation-xml encoding="MathML-Content"> <apply xref="EE"> <and xref="EE.2"/> <apply xref="EE.1"> <xor xref="EE.1.3"/><ci xref="EE.1.2">a</ci><ci xref="EE.1.4">b</ci> </apply> <apply xref="EE.3"> <xor xref="EE.3.3"/><ci xref="EE.3.2">c</ci><ci xref="EE.3.4">d</ci> </apply> </apply> </annotation-xml> <annotation-xml encoding="application/openmath+xml"> <om:OMA xmlns:om="http://www.openmath.org/OpenMath" href="EE"> <om:OMS name="and" cd="logic1" href="EE.2"/> <om:OMA href="EE.1"> <om:OMS name="xor" cd="logic1" href="EE.1.3"/> <om:OMV name="a" href="EE.1.2"/> <om:OMV name="b" href="EE.1.4"/> </om:OMA> <om:OMA href="EE.3"> <om:OMS name="xor" cd="logic1" href="EE.3.3"/> <om:OMV name="c" href="EE.3.2"/> <om:OMV name="d" href="EE.3.4"/> </om:OMA> </om:OMA> </annotation-xml> </semantics> |
Here OMA
, OMS
and OMV
are elements defined in the OpenMath
standard for representing application, symbol, and variable, respectively.
The references from the OpenMath annotation are given by the
href
attributes.
As noted above, the use of namespaces other than MathML, SVG
or HTML within annotation-xml
is not considered valid in the HTML syntax.
Use of colons and namespace-prefixed element names should be avoided
as the HTML parser will generate nodes with local name om:OMA
(for example), and such nodes can not be constructed by a namespace-aware XML parser.
To be effective, MathML must work well with a wide variety of renderers, processors, translators and editors. This chapter raises 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 relate to embedding MathML in [HTML5], and [XHTML], and in any newer HTML when it appears.
There are three kinds of interface issues that arise in embedding MathML in other XML documents. First, MathML markup must be recognized as valid embedded XML content, and not as an error. This issue could be seen primarily as a question of managing namespaces in XML [Namespaces].
Second, in the case of HTML/XHTML, MathML rendering must be integrated with 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 XML content by third-party software or other built-in technology. Examples of this built-in technology are the sophisticated CSS rendering engines now available, and the powerful implementations of JavaScript/ECMAScript that are becoming common. Using these browser-specific mechanisms generally requires additional interface markup of some sort to activate them. In the case of CSS, there is a special restricted form of MathML3 [MathMLforCSS] that is tailored for use with CSS rendering engines that support CSS 2.1 [CSS21]. This restricted profile of MathML3 does not offer the full expressiveness of MathML3, but it provides a portable simpler form that can be rendered acceptably on the screen by modern CSS engines.
Third, other tools for generating and processing MathML must be able to communicate. 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 made to ensure that MathML can easily be generated by user-friendly conversion and authoring tools, and that these tools work together in a dependable, platform-independent, and vendor-independent way.
This chapter applies to both content and presentation markup, and describes
a particular processing model for the semantics
, annotation
and annotation-xml
elements described in
Section 5.1 Annotation Framework.
Within an XML document supporting namespaces [XML],
[Namespaces], the preferred method to recognize
MathML markup is by the identification of the math
element
in the MathML namespace by the use of the MathML namespace
URI http://www.w3.org/1998/Math/MathML
.
The MathML namespace URI is the recommended method to embed MathML within [XHTML] documents. However, some user-agents may require supplementary information to be available to allow them to invoke specific extensions to process the MathML markup.
Markup-language specifications that wish to embed MathML may require special conditions to recognize MathML markup that are independent of this recommendation. The conditions should be similar to those expressed in this recommendation, and the local names of the MathML elements should remain the same as those defined in this recommendation.
HTML does not allow arbitrary namespaces, but has built in knowledge of the MathML namespace.
The math
element and its descendants will be placed in the http://www.w3.org/1998/Math/MathML
namespace by the HTML parser, and will appear to applications as if the input had been XHTML with the namespace declared
as in the previous section. See Section 6.4.3 Mixing MathML and HTML for detailed rules of the HTML parser's handling of MathML.
Although rendering MathML expressions often takes 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. It is important therefore to specify the encoding names by which MathML fragments should be identified.
Outside of those environments where XML namespaces are recognized, media types [RFC2045], [RFC2046] should be used if possible to ensure the invocation of a MathML processor. For those environments where media types are not appropriate, such as clipboard formats on some platforms, the encoding names described in the next section should be used.
MathML contains two distinct vocabularies: one for encoding visual presentation, defined in Chapter 3 Presentation Markup, and one for encoding computational structure, defined in Chapter 4 Content Markup. Some MathML applications may import and export only one of these two vocabularies, while others may produce and consume each in a different way, and still others may process both without any distinction between the two. The following encoding names may be used to distinguish between content and presentation MathML markup when needed.
MathML-Presentation: The instance contains presentation MathML markup only.
Media Type: application/mathml-presentation+xml
Windows Clipboard Flavor: MathML Presentation
Universal Type Identifier: public.mathml.presentation
MathML-Content: The instance contains content MathML markup only.
Media Type: application/mathml-content+xml
Windows Clipboard Flavor: MathML Content
Universal Type Identifier: public.mathml.content
MathML (generic): The instance may contain presentation MathML markup, content MathML markup, or a mixture of the two.
File name extension: .mml
Media Type: application/mathml+xml
Windows Clipboard Flavor: MathML
Universal Type Identifier: public.mathml
See Appendix B Media Types Registrations for more details about each of these encoding names.
MathML 2 specified the predefined encoding values MathML
,
MathML-Content
, and MathML-Presentation
for the
encoding
attribute on the annotation-xml
element.
These values may be used as an alternative to the media type for backward
compatibility. See Section 5.1.3 Alternate representations and
Section 5.1.4 Content equivalents for details.
Moreover, MathML 1.0 suggested the media-type text/mathml
,
which has been superseded by [RFC3023].
MathML expressions are often exchanged between applications using the familiar copy-and-paste or drag-and-drop paradigms and are often stored in files or exchanged over the HTTP protocol. This section provides recommended ways to process MathML during these transfers.
The transfers of MathML fragments described in this section occur between the contexts of two applications by making the MathML data available in several flavors, often called media types, clipboard formats, or data flavors. These flavors are typically ordered by preference by the producing application, and are typically examined in preference order by the consuming application. The copy-and-paste paradigm allows an application to place content in a central clipboard, with one data stream per clipboard format; a consuming application negotiates by choosing to read the data of the format it prefers. The drag-and-drop paradigm allows an application to offer content by declaring the available formats; a potential recipient accepts or rejects a drop based on the list of available formats, and the drop action allows the receiving application to request the delivery of the data in one of the indicated formats. An HTTP GET transfer, as in [HTTP11], allows a client to submit a list of acceptable media types; the server then delivers the data using the one of the indicated media types. An HTTP POST transfer, as in [HTTP11], allows a client to submit data labelled with a media type that is acceptable to the server application.
Current desktop platforms offer copy-and-paste and drag-and-drop
transfers using similar architectures, but with varying naming schemes
depending on the platform. HTTP transfers are all based on media types.
This section specifies what transfer types applications should provide,
how they should be named, and how they should handle the special
semantics
, annotation
, and annotation-xml
elements.
To summarize the three negotiation mechanisms, the following paragraphs will describe transfer flavors, each with a name (a character string) and content (a stream of binary data), which are offered, accepted, and/or exported.
The names listed in Section 6.2.4 Names of MathML Encodings are the exact strings that should be used to identify the transfer flavors that correspond to the MathML encodings. On operating systems that allow such, an application should register their support for these flavor names (e.g. on Windows, a call to RegisterClipboardFormat, or, on the Macintosh platform, declaration of support for the universal type identifier in the application descriptor).
When transferring MathML, an application MUST ensure the content of the data transfer is a well-formed XML instance of a MathML document type. Specifically:
The instance MAY begin with an XML declaration, e.g. <?xml version="1.0">
The instance MUST contain exactly one root math
element.
The instance MUST declare the MathML namespace
on the root math
element.
The instance MAY use a schemaLocation
attribute
on the math
element to indicate the location of the MathML
schema that describes the MathML document type to which the instance
conforms. The presence of the schemaLocation
attribute
does not require a consumer of the MathML instance to obtain or use
the referenced schema.
The instance SHOULD use numeric character references (e.g. α) rather than character entity names (e.g. α) for greater interoperability.
The instance MUST specify the character encoding, if it uses an encoding other than UTF-8, either in the XML declaration, or by the use of a byte-order mark (BOM) for UTF-16-encoded data.
An application that transfers MathML markup SHOULD adhere to the following conventions:
An application that supports pure presentation markup and/or pure content markup SHOULD offer as many of these flavors as it has available.
An application that only exports one MathML flavor SHOULD name it
MathML
if it is unable to determine a more specific flavor.
If an application is able to determine a more specific flavor, it SHOULD
offer both the generic and specific transfer flavors, but it SHOULD only
deliver the specific flavor if it knows that the recipient supports it.
For an HTTP GET transfer, for example, the specific transfer types for
content and presentation markup should only be returned if they are
included in the the HTTP Accept
header sent by the client.
An application that exports the two specific transfer flavors SHOULD
export both the content and presentation transfer flavors, as well as
the generic flavor, which SHOULD combine the other two flavors using
a top-level MathML semantics
element
(see Section 5.4.1 Top-level Parallel Markup).
When an application exports a MathML fragment whose only child of the
root element is a semantics
element, it SHOULD offer, after
the above flavors, a transfer flavor for each annotation
or
annotation-xml
element, provided the transfer flavor can be
recognized and named based on the encoding
attribute value,
and provided the annotation key is (the default)
alternate-representation.
The transfer content for each annotation should contain the character data
in the specified encoding (for an annotation
element), or a
well-formed XML fragment (for an annotation-xml
element), or
the data that results by requesting the URL given by the src
attribute (for an annotation reference).
As a final fallback, an application MAY export a version of
the data in a plain-text flavor (such as text/plain
,
CF_UNICODETEXT
, UnicodeText
, or
NSStringPboardType
). When an application has multiple
versions of an expression available, it may choose the version to
export as text at its discretion. Since some older MathML processors
expect MathML instances transferred as plain text to begin with a
math
element, the text version SHOULD generally omit the XML
declaration, DOCTYPE declaration, and other XML prolog material that
would appear before the math
element. The Unicode
text version of the data SHOULD always be the last flavor exported,
following the principle that exported flavors should be ordered with
the most specific flavor first and the least specific flavor last.
To determine whether a MathML instance is pure content markup or
pure presentation markup, the math
, semantics
,
annotation
and annotation-xml
elements should be
regarded as belonging to both the presentation and content markup
vocabularies. The math
element is treated in this way
because it is required as the root element in any MathML transfer.
The semantics
element and its child annotation elements
comprise an arbitrary annotation mechanism within MathML, and are
not tied to either presentation or content markup. Consequently,
an application that consumes MathML should always process these four
elements, even if it only implements one of the two vocabularies.
It is worth noting that the above recommendations allow agents
that produce MathML to provide binary data for the clipboard, for
example in an image or other application-specific format. The sole
method to do so is to reference the binary data using the src
attribute of an annotation, since XML character data does not allow
for the transfer of arbitrary byte-stream data.
While the above recommendations are intended to improve interoperability between MathML-aware applications that use these transfer paradigms, it should be noted that they do not guarantee interoperability. For example, references to external resources (e.g. stylesheets, etc.) in MathML data can cause interoperability problems if the consumer of the data is unable to locate them, as can happen when cutting and pasting HTML or other data types. An application that makes use of references to external resources is encouraged to make users aware of potential problems and provide alternate ways to obtain the referenced resources. In general, consumers of MathML data that contains references they cannot resolve or do not understand should ignore the external references.
An e-Learning application has a database of quiz questions, some of which contain MathML. The MathML comes from multiple sources, and the e-Learning application merely passes the data on for display, but does not have sophisticated MathML analysis capabilities. Consequently, the application is not aware whether a given MathML instance is pure presentation or pure content markup, nor does it know whether the instance is valid with respect to a particular version of the MathML schema. It therefore places the following data formats on the clipboard:
Flavor Name | Flavor Content |
---|---|
MathML |
<math xmlns="http://www.w3.org/1998/Math/MathML">...</math> |
Unicode Text |
<math xmlns="http://www.w3.org/1998/Math/MathML">...</math> |
An equation editor on the Windows platform is able to generate pure presentation markup, valid with respect to MathML 3. Consequently, it exports the following flavors:
Flavor Name | Flavor Content |
---|---|
MathML Presentation |
<math xmlns="http://www.w3.org/1998/Math/MathML">...</math> |
Tiff |
(a rendering sample) |
Unicode Text |
<math xmlns="http://www.w3.org/1998/Math/MathML">...</math> |
A schema-based content management system on the Mac OS X platform contains multiple MathML representations of a collection of mathematical expressions, including mixed markup from authors, pure content markup for interfacing to symbolic computation engines, and pure presentation markup for print publication. Due to the system's use of schemata, markup is stored with a namespace prefix. The system therefore can transfer the following data:
Flavor Name | Flavor Content |
---|---|
public.mathml.presentation |
<math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation= "http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd"> <mrow> ... <mrow> </math> |
public.mathml.content |
<math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation= "http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd"> <apply> ... <apply> </math> |
public.mathml |
<math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation= "http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd"> <mrow> <apply> ... content markup within presentation markup ... </apply> ... </mrow> </math> |
public.plain-text.tex |
{x \over x-1} |
public.plain-text |
<math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation= "http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd"> <mrow> ... <mrow> </math> |
A similar content management system is web-based and delivers MathML representations of mathematical expressions. The system is able to produce MathML-Presentation, MathML-Content, TeX and pictures in TIFF format. In web-pages being browsed, it could produce a MathML fragment such as the following:
<math xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow>...</mrow> <annotation-xml encoding="MathML-Content">...</annotation-xml> <annotation encoding="TeX">{1 \over x}</annotation> <annotation encoding="image/tiff" src="formula3848.tiff"/> </semantics> </math>
A web-browser on the Windows platform that receives such a fragment and tries to export it as part of a drag-and-drop action, can offer the following flavors:
Flavor Name | Flavor Content |
---|---|
MathML Presentation |
<math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation= "http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd"> <mrow> ... <mrow> </math> |
MathML Content |
<math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation= "http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd"> <apply> ... <apply> </math> |
MathML |
<math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation= "http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd"> <mrow> <apply> ... content markup within presentation markup ... </apply> ... </mrow> </math> |
TeX |
{x \over x-1} |
CF_TIFF |
(the content of the picture file, requested from formula3848.tiff) |
CF_UNICODETEXT |
<math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation= "http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd"> <mrow> ... <mrow> </math> |
MathML is usually used in combination with other markup languages. The most typical case is perhaps the use of MathML within a document-level markup language, such as HTML or DocBook. It is also common that other object-level markup languages are also included in a compound document format, such as MathML and SVG in HTML5. Other common use cases include mixing other markup within MathML. For example, an authoring tool might insert an element representing a cursor position or other state information within MathML markup, so that an author can pick up editing where it was broken off.
Most document markup languages have some concept of an inline
equation, (or graphic, object, etc.) so there is a typically a natural
way to incorporate MathML instances into the content
model. However, in the other direction, embedding of markup within
MathML is not so clear cut, since in many MathML elements, the role of
child elements is defined by position. For example, the first
child of an apply
must be an operator, and the second child
of an mfrac
is the denominator. The proper behavior when
foreign markup appears in such contexts is problematic. Even when such
behavior can be defined in a particular context, it presents an
implementation challenge for generic MathML processors.
For this reason, the default MathML schema does not allow foreign markup elements to be included within MathML instances.
In the standard schema, elements from other namespaces are not allowed, but attributes from other namespaces are permitted. MathML processors that encounter unknown XML markup should behave as follows:
An attribute from a non-MathML namespace should be silently ignored.
An element from a non-MathML namespace should be treated
as an error, except in an annotation-xml
element.
If the element is a child of a presentation element, it should be
handled as described in Section 3.3.5 Error Message <merror>
.
If the element is a child of a content element, it should be
handled as described in Section 4.2.9 Error Markup <cerror>
.
For example, if the second child of an mfrac
element is an
unknown element, the fraction should be rendered with a denominator
that indicates the error.
When designing a compound document format in which MathML is included in a larger document type, the designer may extend the content model of MathML to allow additional elements. For example, a common extension is to extend the MathML schema such that elements from non-MathML namespaces are allowed in token elements, but not in other elements. MathML processors that encounter unknown markup should behave as follows:
An unrecognized XML attribute should be silently ignored.
An unrecognized element in a MathML token element should be silently ignored.
An element from a non-MathML namespace should be treated
as an error, except in an annotation-xml
element.
If the element is a child of a presentation element, it should be
handled as described in Section 3.3.5 Error Message <merror>
.
If the element is a child of a content element, it should be
handled as described in Section 4.2.9 Error Markup <cerror>
.
Extending the schema in this way is easily achieved using the Relax NG schema described in Appendix A Parsing MathML, it may be as simple as including the MathML schema whilst overriding the content model of mtext
:
default namespace m = "http://www.w3.org/1998/Math/MathML" include "mathml3.rnc" { mtext = element mtext {mtext.attributes, (token.content|anyElement)*} }
The definition given here would allow any well formed XML that is not in the MathML namespace as a child of mtext
. In practice this may be too lax. For example, an XHTML+MathML Schema may just want to allow inline XHTML elements as additional
children of mtext
. This may be achieved by replacing anyElement
by a suitable production from the schema for the host document type, see Section 6.4.1 Mixing MathML and XHTML.
Considerations about mixing markup vocabularies in compound documents arise when a compound document type is first designed. But once the document type is fixed, it is not generally practical for specific software tools to further modify the content model to suit their needs. However, it is still frequently the case that such tools may need to store additional information within a MathML instance. Since MathML is most often generated by authoring tools, a particularly common and important case is where an authoring tool needs to store information about its internal state along with a MathML expression, so an author can resume editing from a previous state. For example, placeholders may be used to indicate incomplete parts of an expression, or a insertion point within an expression may need to be stored.
An application that needs to persist private data within a MathML expression should generally attempt to do so without altering the underlying content model, even in situations where it is feasible to do so. To support this requirement, regardless of what may be allowed by the content model of a particular compound document format, MathML permits the storage of private data via the following strategies:
In a format that permits the use of XML Namespaces, for small amounts of data, attributes from other namespaces are allowed on all MathML elements.
For larger amounts of data, applications may use the
semantics
element, as described in
Section 5.1 Annotation Framework.
For authoring tools and other applications that need to
associate particular actions with presentation MathML subtrees,
e.g. to mark an incomplete expression to be filled in by an author,
the maction
element may be used, as described in
Section 3.7.1 Bind Action to Sub-Expression
<maction>
.
To fully integrate MathML into XHTML, it should be possible not only to embed MathML in
XHTML, but also to embed XHTML in MathML.
The schema used for the W3C HTML5 validator extends mtext
to allow all
inline (phrasing) HTML elements (including svg
) to be used within the
content of mtext
. See the example in Section 3.2.2.2 Embedding HTML in MathML. As noted above,
MathML fragments using XHTML elements within mtext
will not be valid MathML if extracted
from the document and used in isolation. Editing tools may offer support for removing any HTML
markup from within mtext
and replacing it by a text alternative.
In most cases, XHTML elements (headings, paragraphs, lists, etc.) either do not apply in mathematical contexts, or MathML already provides equivalent or improved functionality specifically tailored to mathematical content (tables, mathematics style changes, etc.).
Consult the W3C Math Working Group home page for compatibility and implementation suggestions for current browsers and other MathML-aware tools.
There may be non-XML vocabularies which require markup for mathematical expressions,
where it makes sense to reference this specification. HTML is an important example discussed in the
next section, however other examples exist. It is possible to use a TeX-like syntax such as
\frac{a}{b}
rather than explicitly using <mfrac>
and <mi">
. If a system parses a specified syntax and produces a tree that may be
validated against the MathML schema then it may be viewed as
as a MathML application. Note however that documents using such a system are not valid MathML.
Implementations of such a syntax should, if possible, offer a facility to
output any mathematical expressions as MathML in the XML syntax defined here. Such an application
would then be a MathML-output-conformant processor as described in Section 2.3.1 MathML Conformance.
An important example of a non-XML based system is defined in [HTML5]. When
considering MathML in HTML there are two separate issues to consider. Firstly the schema is extended
to allow HTML in mtext
as described above in the context of XHTML. Secondly an HTML parser
is used rather than an XML parser. The parsing of MathML by an HTML parser is normatively defined in
[HTML5]. The description there is aimed at parser implementers and written in terms of
the state transitions of the parser as it parses each character of the input. The
non-normative description below aims to give a higher level description and
examples.
XML parsing is completely regular, any XML document may be parsed without reference to the particular vocabulary being used. HTML parsing differs in that it is a custom parser for the HTML vocabulary with specific rules for each element. Similarly to XML though, the HTML parser distinguishes parsing from validation; some input, even if it renders correctly, is classed as a parse error which may be reported by validators (but typically is not reported by rendering systems).
The main differences that affect MathML usage may be summarized as:
Attribute values in most cases do not need to be quoted: <mfenced open=(
close=)>
would parse correctly.
End tags may in many cases be omitted.
HTML does not support namespaces other than the three built in ones for HTML, MathML and SVG, and
does not support namespace prefixes. Thus you can not use a prefix form like <mml:math
xmlns:mml="http://www.w3.org/1998/Math/MathML">
and while you may use <math
xmlns="http://www.w3.org/1998/Math/MathML">
, the namespace declaration is essentially ignored
and the input is treated as <math>
. In either case the math
element and its
descendants are placed in the MathML namespace. As noted in Chapter 5 Mixing Markup Languages for Mathematical Expressions the lack of
namespace support limits some of the possibilities for annotating MathML with markup from other
vocabularies when used in HTML.
Unlike the XML parser, the HTML parser is defined to accept any input string
and produce a defined result (which may be classified as non-conforming. The extreme example
<math></<><z =5>
for example would be flagged as a parse error by validators but would return a tree corresponding to
a math element containing a comment <
and an element z
with an attribute
that could not be expressed in XML with name =5
and value ""
.
Unless inside the token elements <mtext>
, <mo>
, <mn>
, <mi>
,
<ms>
, or inside an <annotation-xml>
with encoding
attribute
"text/html" or "annotation/xhtml+xml", the presence of an HTML element
will terminate the math expression by closing all open MathML elements, so
that the HTML element is interpreted as being in the outer HTML context. Any following MathML
elements are then not contained in <math>
so will be parsed as invalid HTML elements and not
rendered as MathML. See for example the example given in Section 5.2.3.3 Using annotation-xml
in HTML documents.
In the interests of compatibility with existing MathML applications authors and editing systems should use MathML fragments that are well formed XML, even when embedded in an HTML document. Also as noted above, although applications accepting MathML in HTML documents must accept MathML making use of these HTML parser features, they should offer a way to export MathML in a portable XML syntax.
In MathML 3, an element is designated as a link by the presence of
the href
attribute. MathML has no element that corresponds
to the HTML/XHTML anchor element a.
MathML allows the href
attribute on all
elements.
However, most user agents have no way
to implement nested links or links on elements with no visible rendering;
such links may have no effect.
The list of presentation markup elements that do not ordinarily have a visual rendering, and thus should not be used as linking elements, is given in the table below.
MathML elements that should not be linking elements | |
---|---|
mprescripts |
none |
malignmark |
maligngroup |
For compound document formats that support linking mechanisms, the
id
attribute should
be used to specify the location for a link into a MathML expression. The
id
attribute is allowed on all MathML elements, and its value
must be unique within a document, making it ideal for this purpose.
Note that MathML 2 has no direct support for linking; it refers to
the W3C Recommendation "XML Linking Language" [XLink] in
defining links in compound document contexts by using an xlink:href
attribute.
As mentioned above, MathML 3 adds an href
attribute for linking so that xlink:href
is no longer needed.
However, xlink:href
is still allowed because MathML permits the use of attributes
from non-MathML namespaces. It is recommended that new compound document
formats use the MathML 3 href
attribute for linking. When user agents
encounter MathML elements with both href
and xlink:href
attributes, the
href
attribute should take precedence. To support backward
compatibility, user agents that implement XML Linking
in compound documents containing MathML 2 should continue to support
the use of the xlink:href
attribute in addition to supporting the href
attribute.
Apart from the introduction of new glyphs, many of the situations where one might be inclined to use an image amount to displaying labeled diagrams. For example, knot diagrams, Venn diagrams, Dynkin diagrams, Feynman diagrams and commutative diagrams all fall into this category. As such, their content would be better encoded via some combination of structured graphics and MathML markup. However, at the time of this writing, it is beyond the scope of the W3C Math Activity to define a markup language to encode such a general concept as "labeled diagrams." (See http://www.w3.org/Math for current W3C activity in mathematics and http://www.w3.org/Graphics for the W3C graphics activity.)
One mechanism for embedding additional graphical content is via the
semantics
element, as in the following example:
<semantics> <apply> <intersect/> <ci>A</ci> <ci>B</ci> </apply> <annotation-xml encoding="image/svg+xml"> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 290 180"> <clipPath id="a"> <circle cy="90" cx="100" r="60"/> </clipPath> <circle fill="#AAAAAA" cy="90" cx="190" r="60" style="clip-path:url(#a)"/> <circle stroke="black" fill="none" cy="90" cx="100" r="60"/> <circle stroke="black" fill="none" cy="90" cx="190" r="60"/> </svg> </annotation-xml> <annotation-xml encoding="application/xhtml+xml"> <img xmlns="http://www.w3.org/1999/xhtml" src="intersect.gif" alt="A intersect B"/> </annotation-xml> </semantics>
Here, the annotation-xml
elements are used to indicate alternative
representations of the MathML-Content depiction of the
intersection of two sets.
The first one is in the "Scalable Vector
Graphics" format [SVG1.1]
(see [XHTML-MathML-SVG] for the definition of an XHTML profile integrating MathML and SVG), the second one uses the
XHTML img
element embedded as an XHTML fragment.
In this situation, a MathML processor can use any of these
representations for display, perhaps producing a graphical format
such as the image below.
Note that the semantics representation of this example is given
in MathML-Content markup, as the first child of the
semantics
element. In this regard, it is the
representation most analogous to the alt
attribute of the
img
element in XHTML, and would likely be
the best choice for non-visual rendering.
When MathML is rendered in an environment that supports CSS [CSS21], controlling mathematics style properties with a CSS style sheet is desirable, but not as simple as it might first appear, because the formatting of MathML layout schemata is quite different from the CSS visual formatting model and many of the style parameters that affect mathematics layout have no direct textual analogs. Even in cases where there are analogous properties, the sensible values for these properties may not correspond. Because of this difference, applications that support MathML natively may choose to restrict the CSS properties applicable to MathML layout schemata to those properties that do not affect layout.
Generally speaking, the model for CSS interaction with the math style attributes runs as follows. A CSS style sheet might provide a style rule such as:
math *.[mathsize="small"] { font-size: 80% }
This rule sets the CSS font-size property 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 CSS 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 CSS stylesheets for MathML. Because changing typographic properties of mathematics symbols can change the meaning of an equation, stylesheets should be written in a way such that changes to document-wide typographic styles do not affect embedded MathML expressions.
Another pitfall to be avoided is using CSS to provide typographic style information necessary to the proper understanding of an expression. Expressions dependent on CSS for meaning will not be portable to non-CSS environments such as computer algebra systems. By using the logical values of the new MathML 3.0 mathematics style attributes as selectors for CSS rules, it can be assured that style information necessary to the sense of an expression is encoded directly in the MathML.
MathML 3.0 does not specify how a user agent should process style information, because there are many non-CSS MathML environments, and because different users agents and renderers have widely varying degrees of access to CSS information.
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 requested by various sources should occur. The order is defined by [CSS21] cascading order taking into account precedence of non-CSS presentational hints.
Notation and symbols have proved very important for mathematics. Mathematics has grown in part because its notation continually changes toward being succinct and suggestive. Many new signs have been developed for use in mathematical notation, and many have been adopted that were originally introduced elsewhere.The result is that mathematics makes use of a very large collection of symbols. It is difficult to write mathematics fluently if these characters are not available for use. It is difficult to read mathematics if corresponding glyphs are not available for presentation on specific display devices.
The W3C Math Working Group therefore took on the job of specifying part of the mechanism needed to proceed from notation to final presentation, and has collaborated with the Unicode Technical Committee (UTC) and the STIX Fonts Project in undertaking specification of the rest.
This chapter contains discussion of characters for use within MathML, recommendations for their use, and warnings concerning the correct form of the corresponding code points given in the Universal Multiple-Octet Coded Character Set (UCS) [ISO10646] as codified in Unicode [Unicode]. For simplicity we refer to this character set by the short name Unicode. Unless otherwise stated, the mappings discussed in this chapter and elsewhere in the MathML 3.0 recommendation are based on Unicode 5.2. Conformant MathML processors (see Section 2.3 Conformance) are free to use characters defined in Unicode 5.2 or later.
While a long process of review and adoption by UTC and ISO/IEC of the characters of special interest to mathematics and MathML is now complete, more characters may be added in the future. For the latest character tables and font information, see the [Entities] and the Unicode Home Page, notably Unicode Work in Progress and Unicode Technical Report #25 “Unicode Support for Mathematics”.
A MathML token element (see Section 3.2 Token Elements, Section 4.2.1 Numbers <cn>
,
Section 4.2.2 Content Identifiers <ci>
, Section 4.2.3 Content Symbols <csymbol>
) takes as content a
sequence of MathML characters or mglyph
elements. The latter are
used to represent characters that do not have a Unicode encoding, as described in Section 3.2.1.2 Using images to represent
symbols <mglyph/>
. The need for mglyph
should be rare because Unicode 3.1
provided approximately one thousand alphabetic characters for mathematics, and
Unicode 3.2 added over 900 more special mathematical symbols.
Any character allowed by XML may be used in MathML. More precisely, the legal Unicode characters have the hexadecimal code numbers 09 (tab = U+0009), 0A (line feed = U+000A), 0D (carriage return = U+000D), 20-D7FF (U+0020..U+D7FF), E000-FFFD (U+E000..U+FFFD), and 10000-10FFFF (U+10000..U+10FFFF). The exclusions above code number D7FF are of the blocks used in surrogate pairs, and the two characters guaranteed not to be Unicode characters at all. U+FFFE is excluded to allow determination of byte order in certain encodings.
There are essentially three different ways of encoding character data in an XML document.
Using characters directly: For example, the 'é' (character U+00E9 [LATIN SMALL LETTER E WITH ACUTE]) may have been inserted. This option is only useful if the character encoding specified for the XML document includes the character intended. Note that if the document is, for example, encoded in Latin-1 (ISO-8859-1) then only the characters in that encoding are available directly; for instance character U+00E9 (eacute) is, but character U+03B1 (alpha) is not.
Using numeric XML character references: For example, 'é' may be represented as é (decimal) or é (hex), or é (decimal) or é. Note that the numbers in the character references always refer to the Unicode encoding (and not to the character encoding used in the XML file). By using character references it is always possible to access the entire Unicode range.
Using entity references: The MathML DTD defines internal entities that expand to character data. Thus for example the entity reference é may be used rather than the character reference é. An XML fragment that uses an entity reference which is not defined in a DTD is not well-formed; therefore it will be rejected by an XML parser. For this reason every fragment using entity references must use a DOCTYPE declaration which specifies the MathML DTD, or a DTD that at least declares any entity reference used in the MathML instance. The need to use a DOCTYPE complicates inclusion of MathML in some documents. However, entity references can be useful for small illustrative examples.
Earlier versions of this MathML specification included detailed listings of the entity definitions to be used with the MathML DTD. These entity definitions are of more general use, and have now been separated into an ancillary document, XML Entity Definitions for Characters [Entities]. The tables there list the entity names and the corresponding Unicode character references. That document describes several entity sets; not all of them are used in the MathML DTD. The MathML DTD references the combined HTML MathML entity set defined in [Entities].
For special purposes, one may need a symbol which does not have a
Unicode representation.
In these cases one may use the mglyph
element for direct access to a glyph as an image, or (in some systems)
from a font that uses a non-Unicode encoding.
All MathML token elements accept characters in their content and
also accept an mglyph
there.
Beware, however, that use of mglyph
to access a font is
deprecated and the mechanism may not work in all systems. The
mglyph
element should always supply a useful alternative
representation in its alt
attribute.
In mathematical and scientific writing, single letters often denote variables and constants in a given context. The increasing complexity of science has led to the use of certain common alphabet and font variations to provide enough special symbols of this letter-like type. These denotations are generally not letters that may be used to make up words with recognized meanings, but individual carriers of semantics themselves. Writing a string of such symbols is usually interpreted in terms of some composition law, for instance, multiplication. Many letter-like symbols may be quickly interpreted as of a certain mathematical type by specialists in a given area: for instance, bold symbols, whether based on Latin or Greek letters, as vectors in physics or engineering, or Fraktur symbols as Lie algebras in part of pure mathematics.
The additional Mathematical Alphanumeric Symbols provided in Unicode 3.1 have code points in the range U+1D400 to U+1D7FF in Plane 1, that is, in the first plane with Unicode values higher than 216. This plane of characters is also known as the Secondary Multilingual Plane (SMP), in contrast to the Basic Multilingual Plane (BMP) which was originally the entire extent of Unicode. Support for Plane 1 characters in currently deployed software is not always reliable, but it should be possible in multilingual operating systems, since Plane 2 has many Chinese characters that must be displayable in East Asian locales.
As discussed in Section 3.2.2 Mathematics style attributes common to token elements, MathML offers an
alternative mechanism to specify mathematical alphanumeric characters.
This alternative mechanism spans the gap between the
specification of the mathematical alphanumeric symbols as Unicode
code points, and the deployment of software and fonts that support them.
Namely, one uses the mathvariant
attribute on a token element
such as mi
to indicate that the character data in the token
element selects a mathematical alphanumeric symbol.
In principle, any mathvariant value may be used with any character
data to define a specific symbolic token. In practice, only certain
combinations of character data and mathvariant values will be visually
distinguished by a given renderer.
In this section we explain the correspondence between certain characters
in Plane 0 that, when modified by the mathvariant
attribute,
are considered equivalent to mathematical alphanumeric symbol characters.
The mathematical alphanumeric symbol characters in Plane 1 include alphabets for Latin upper-case and lower-case letters, including dotless i and j, Greek upper-case and lower-case letters, Greek symbols (also known as variants), including upper-case and lower-case digamma, and Latin digits. These alphabets provide Plane 1 Unicode code points that differ from corresponding Plane 0 characters only by a variation in font that carries mathematical semantics when used in a formula.
The mathvariant
attribute uses exactly this correspondence to
provide an alternate markup encoding that selects these Plane 1 characters.
For example, the Mathematical Italic alphabet runs from U+1D434 ("A")
to U+1D467 ("z"). Thus, a typical example of an identifier for a
variable, marked up as
<mi>a</mi> |
and rendered in a mathematical italic font (as described in
Section 3.2.3 Identifier <mi>
) could equivalently be marked up as
<mi>𝑎<!--MATHEMATICAL ITALIC SMALL A--><!--MATHEMATICAL ITALIC SMALL A--></mi> |
which invokes the Mathematical Italic lower-case a explicitly.
An important use of the mathematical alphanumeric symbols in Plane 1 is for identifiers normally printed in special mathematical fonts, such as Fraktur, Greek, Boldface, or Script. As another example, the Mathematical Fraktur alphabet runs from U+1D504 ("A") to U+1D537 ("z"). Thus, an identifier for a variable that uses Fraktur characters could be marked up as
<mi>𝔄<!--MATHEMATICAL FRAKTUR CAPITAL A--><!--BLACK-LETTER CAPITAL A--></mi> |
An alternative, equivalent markup for this example is to use the
common upper-case A, modified by using the mathvariant
attribute:
<mi mathvariant="fraktur">A</mi> |
A MathML processor must treat a mathematical alphanumeric character (when it appears) as identical to the corresponding combination of the unstyled character and mathvariant attribute value. It is important to note that the mathvariant attribute specifies a semantic class of characters, each of which has a specific appearance that should be protected from document-wide style changes, so the intended meaning of the character may be preserved. The use of a mathematical alphanumeric character is also intended to preserve this specific appearance, and so these characters are also not to be affected by surrounding style changes.
Not all combinations of character data and mathvariant values have assigned Unicode code points. For example, sans-serif Greek alphabets are omitted, while bold sans-serif Greek alphabets are included, and bold digits are included, while bold-italic digits are excluded. A renderer should visually distinguish those combinations of character data and mathvariant attribute values that it can subject to the availability of font characters. It is intended that renderers distinghish at least those combinations that have equivalent Unicode code points, and renderers are free to ignore those combinations that have no assigned Unicode code point or for which adequate font support is unavailable.
The exact correspondence between a mathematical alphabetic character and an unstyled character is complicated by the fact that certain characters that were already present in Unicode in Plane 0 are not in the 'expected' sequence in Plane 1. The table below shows the Plane 0 mathematical alphanumeric symbols, listing for each character its Unicode code point, its Unicode character name, its corresponding unstyled alphabetic character, and the code point in Plane 1 where one might naturally have sought this character.
Unicode code point | Unicode name | BMP code | Plane-1 code |
---|---|---|---|
U+210E | PLANCK CONSTANT | U+0068 | U+1D455 |
U+212C | SCRIPT CAPITAL B | U+0042 | U+1D49D |
U+2130 | SCRIPT CAPITAL E | U+0045 | U+1D4A0 |
U+2131 | SCRIPT CAPITAL F | U+0046 | U+1D4A1 |
U+210B | SCRIPT CAPITAL H | U+0048 | U+1D4A3 |
U+2110 | SCRIPT CAPITAL I | U+0049 | U+1D4A4 |
U+2112 | SCRIPT CAPITAL L | U+004C | U+1D4A7 |
U+2133 | SCRIPT CAPITAL M | U+004D | U+1D4A8 |
U+211B | SCRIPT CAPITAL R | U+0052 | U+1D4AD |
U+212F | SCRIPT SMALL E | U+0065 | U+1D4BA |
U+210A | SCRIPT SMALL G | U+0067 | U+1D4BC |
U+2134 | SCRIPT SMALL O | U+006F | U+1D4C4 |
U+212D | BLACK-LETTER CAPITAL C | U+0043 | U+1D506 |
U+210C | BLACK-LETTER CAPITAL H | U+0048 | U+1D50B |
U+2111 | BLACK-LETTER CAPITAL I | U+0049 | U+1D50C |
U+211C | BLACK-LETTER CAPITAL R | U+0052 | U+1D515 |
U+2128 | BLACK-LETTER CAPITAL Z | U+005A | U+1D51D |
U+2102 | DOUBLE-STRUCK CAPITAL C | U+0043 | U+1D53A |
U+210D | DOUBLE-STRUCK CAPITAL H | U+0048 | U+1D53F |
U+2115 | DOUBLE-STRUCK CAPITAL N | U+004E | U+1D545 |
U+2119 | DOUBLE-STRUCK CAPITAL P | U+0050 | U+1D547 |
U+211A | DOUBLE-STRUCK CAPITAL Q | U+0051 | U+1D548 |
U+211D | DOUBLE-STRUCK CAPITAL R | U+0052 | U+1D549 |
U+2124 | DOUBLE-STRUCK CAPITAL Z | U+005A | U+1D551 |
Mathematical Alphanumeric Symbol characters should not be used for styled prose. For example, Mathematical Fraktur A must not be used to just select a blackletter font for an uppercase A as it would create problems for searching, restyling (e.g. for accessibility), and many other kinds of processing.
Some characters, although important for the quality of print or alternative rendering, do not have glyph marks that correspond directly to them. They are called here non-marking characters. Their roles are discussed in Chapter 3 Presentation Markup and Chapter 4 Content Markup.
In MathML, control of page composition, such as line-breaking, is
effected by the use of the proper attributes on the mo
and mspace
elements.
The characters below are not simple spacers. They are especially important new additions to the UCS because they provide textual clues which can increase the quality of print rendering, permit correct audio rendering, and allow the unique recovery of mathematical semantics from text which is visually ambiguous.
Unicode code point | Unicode name | Description |
---|---|---|
U+2061 | FUNCTION APPLICATION | character showing function application in presentation tagging
(Section 3.2.5 Operator, Fence, Separator or Accent
<mo> )
|
U+2062 | INVISIBLE TIMES | marks multiplication when it is understood without a mark
(Section 3.2.5 Operator, Fence, Separator or Accent
<mo> )
|
U+2063 | INVISIBLE SEPARATOR | used as a separator, e.g., in indices (Section 3.2.5 Operator, Fence, Separator or Accent
<mo> )
|
U+2064 | INVISIBLE PLUS | marks addition, especially in constructs such as 1½
(Section 3.2.5 Operator, Fence, Separator or Accent
<mo> )
|
Some characters which occur fairly often in mathematical texts, and have special significance there, are frequently confused with other similar characters in the UCS. In some cases, common keyboard characters have become entrenched as alternatives to the more appropriate mathematical characters. In others, characters have legitimate uses in both formulas and text, but conflicting rendering and font conventions. All these characters are called here anomalous characters.
Typical Latin-1-based keyboards contain several characters that are visually similar to important mathematical characters. Consequently, these characters are frequently substituted, intentionally or unintentionally, for their more correct mathematical counterparts.
The most common ordinary text character which enjoys a special
mathematical use is U+002D [HYPHEN-MINUS]. As its Unicode name
suggests, it is used as a hyphen in prose contexts, and as a minus
or negative sign in formulas.
For text use, there is a specific code point U+2010 [HYPHEN] which is
intended for prose contexts, and which should render as a hyphen or
short dash.
For mathematical use, there is another code point U+2212 [MINUS SIGN]
which is intended for mathematical formulas, and which should render
as a longer minus or negative sign.
MathML renderers should treat U+002D [HYPHEN-MINUS] as equivalent to
U+2212 [MINUS SIGN] in formula contexts such as mo
, and as
equivalent to U+2010 [HYPHEN] in text contexts such as mtext
.
On a typical European keyboard there is a key available which is viewed as an apostrophe or a single quotation mark (an upright or right quotation mark). Thus one key is doing double duty for prose input to enter U+0027 [APOSTROPHE] and U+2019 [RIGHT SINGLE QUOTATION MARK]. In mathematical contexts it is also commonly used for the prime, which should be U+2032 [PRIME]. Unicode recognizes the overloading of this symbol and remarks that it can also signify the units of minutes or feet. In the unstructured printed text of normal prose the characters are placed next to one another. The U+0027 [APOSTROPHE] and U+2019 [RIGHT SINGLE QUOTATION MARK] are marked with glyphs that are small and raised with respect to the center line of the text. The fonts used provide small raised glyphs in the appropriate places indexed by the Unicode codes. The U+2032 [PRIME] of mathematics is similarly treated in fuller Unicode fonts.
MathML renderers are encouraged to treat U+0027 [APOSTROPHE] as U+2032 [PRIME] when appropriate in formula contexts.
A final remark is that a ‘prime’ is often used in transliteration of the Cyrillic character U+044C [CYRILLIC SMALL LETTER SOFT SIGN]. This different use of primes is not part of considerations for mathematical formulas.
While the minus and prime characters are the most common and important keyboard characters with more precise mathematical counterparts, there are a number of other keyboard character substitutions that are sometime used. For example some may expect
<mo>''</mo> |
to be treated as U+2033 [DOUBLE PRIME], and analogous substitutions could perhaps be made for U+2034 [TRIPLE PRIME] and U+2057 [QUADRUPLE PRIME]. Similarly, sometimes U+007C [VERTICAL LINE] is used for U+2223 [DIVIDES]. MathML regards these as application-specific authoring conventions, and recommends that authoring tools generate markup using the more precise mathematical characters for better interoperability.
There are a number of characters in the UCS that traditionally have been taken to have a natural ‘script’ aspect. The visual presentation of these characters is similar to a script, that is, raised from the baseline, and smaller than the base font size. The degree symbol and prime characters are examples. For use in text, such characters occur in sequence with the identifier they follow, and are typically rendered using the same font. These characters are called pseudo-scripts here.
In almost all mathematical contexts, pseudo-script characters should be associated with a base expression using explicit script markup in MathML. For example, the preferred encoding of "x prime" is
<msup><mi>x</mi><mo>′<!--PRIME--><!--PRIME--></mo></msup> |
and not
<mi>x'</mi> |
or any other variants not using an explicit script construct. Note, however, that within
text contexts such as mtext
, pseudo-scripts may be used in sequence with other character data.
There are two reasons why explicit markup is preferable in mathematical contexts. First, a problem arises with typesetting, when pseudo-scripts are used with subscripted identifiers. Traditionally, subscripting of x' would be rendered stacked under the prime. This is easily accomplished with script markup, for example:
<mrow><msubsup><mi>x</mi><mn>0</mn><mo>′<!--PRIME--><!--PRIME--></mo></msubsup></mrow> |
By contrast,
<mrow><msub><mi>x'</mi><mn>0</mn></msub></mrow> |
will render with staggered scripts.
Note this means that a renderer of MathML will have to treat pseudo-scripts differently from most other character codes it finds in a superscript position; in most fonts, the glyphs for pseudo-scripts are already shrunk and raised from the baseline.
The second reason that explicit script markup is preferrable to juxtaposition of characters is that it generally better reflects the intended mathematical structure. For example,
<msup> <mrow><mo>(</mo><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow><mo>)</mo></mrow> <mo>′<!--PRIME--><!--PRIME--></mo> </msup> |
accurately reflects that the prime here is operating on an entire expression, and does not suggest that the prime is acting on the final right parenthesis.
However, the data model for all MathML token elements is Unicode text, so one cannot rule out the possibility of valid MathML markup containing constructions such as
<mrow><mi>x'</mi></mrow> |
and
<mrow><mi>x</mi><mo>'</mo></mrow> |
While the first form may, in some rare situations, legitmately be used to distinguish a multi-character identifer named x' from the derivative of a function x, such forms should generally be avoided. Authoring and validation tools are encouraged to generate the recommended script markup:
<mrow><msup><mi>x</mi><mo>′<!--PRIME--><!--PRIME--></mo></msup></mrow> |
The U+2032 [PRIME] character is perhaps the most common pseudo-script, but there are many others, as listed below:
Pseudo-script Characters | |
---|---|
U+0022 | QUOTATION MARK |
U+0027 | APOSTROPHE |
U+002A | ASTERISK |
U+0060 | GRAVE ACCENT |
U+00AA | FEMININE ORDINAL INDICATOR |
U+00B0 | DEGREE SIGN |
U+00B2 | SUPERSCRIPT TWO |
U+00B3 | SUPERSCRIPT THREE |
U+00B4 | ACUTE ACCENT |
U+00B9 | SUPERSCRIPT ONE |
U+00BA | MASCULINE ORDINAL INDICATOR |
U+2018 | LEFT SINGLE QUOTATION MARK |
U+2019 | RIGHT SINGLE QUOTATION MARK |
U+201A | SINGLE LOW-9 QUOTATION MARK |
U+201B | SINGLE HIGH-REVERSED-9 QUOTATION MARK |
U+201C | LEFT DOUBLE QUOTATION MARK |
U+201D | RIGHT DOUBLE QUOTATION MARK |
U+201E | DOUBLE LOW-9 QUOTATION MARK |
U+201F | DOUBLE HIGH-REVERSED-9 QUOTATION MARK |
U+2032 | PRIME |
U+2033 | DOUBLE PRIME |
U+2034 | TRIPLE PRIME |
U+2035 | REVERSED PRIME |
U+2036 | REVERSED DOUBLE PRIME |
U+2037 | REVERSED TRIPLE PRIME |
U+2057 | QUADRUPLE PRIME |
In addition, the characters in the Unicode Superscript and Subscript block (beginning at U+2070) should be treated as pseudo-scripts when they appear in mathematical formulas.
Note that several of these characters are common on keyboards, including U+002A [ASTERISK], U+00B0 [DEGREE SIGN], U+2033 [DOUBLE PRIME], and U+2035 [REVERSED PRIME] also known as a back prime.
In the UCS there are many combining characters that are intended to be used for the many accents of numerous different natural languages. Some of them may seem to provide markup needed for mathematical accents. They should not be used in mathematical markup. Superscript, subscript, underscript, and overscript constructions as just discussed above should be used for this purpose. Of course, combining characters may be used in multi-character identifiers as they are needed, or in text contexts.
There is one more case where combining characters turn up naturally in mathematical markup. Some relations have associated negations, such as U+226F [NOT GREATER-THAN] for the negation of U+003E [GREATER-THAN SIGN]. The glyph for U+226F [NOT GREATER-THAN] is usually just that for U+003E [GREATER-THAN SIGN] with a slash through it. Thus it could also be expressed by U+003E-0338 making use of the combining slash U+0338 [COMBINING LONG SOLIDUS OVERLAY]. That is true of 25 other characters in common enough mathematical use to merit their own Unicode code points. In the other direction there are 31 character entity names listed in [Entities] which are to be expressed using U+0338 [COMBINING LONG SOLIDUS OVERLAY].
In a similar way there are mathematical characters which have negations given by a vertical bar overlay U+20D2 [COMBINING LONG VERTICAL LINE OVERLAY]. Some are available in pre-composed forms, and some named character entities are given explicitly as combinations. In addition there are examples using U+0333 [COMBINING DOUBLE LOW LINE] and U+20E5 [COMBINING REVERSE SOLIDUS OVERLAY], and variants specified by use of the U+FE00 [VARIATION SELECTOR-1]. For fuller listing of these cases see the listings in [Entities].
The general rule is that a base character followed by a string of combining characters should be treated just as though it were the pre-composed character that results from the combination, if such a character exists.
A MathML document must be a well-formed XML document using elements in the MathML namespace as defined by this specification, however it is not required that the document refer to any specific Document Type Definition (DTD) or schema that specifies MathML. It is sometimes advantageous not to specify such a language definition as these files are large, often much larger than the MathML expression and unless they have been previously cached by the MathML application, the time taken to fetch the DTD or schema may have an appreciable effect on the processing of the MathML document.
Note that if no DTD is specified with a DOCTYPE declaration, that entity references (for example to refer to MathML characters by name) may not be used. The document should be encoded in an encoding (for example UTF-8) in which all needed characters may be encoded as character data, or characters may be referenced using numeric character references, for example ∫ rather than ∫
If a MathML fragment is parsed without a DTD, in other words as a well-formed XML fragment, it is the responsibility of the processing application to treat the white space characters occurring outside of token elements as not significant.
However, in many circumstances, especially while producing or editing MathML, it is useful to use a language definition to constrain the editing process or to check the correctness of generated files. The following section, Section A.2 Using the RelaxNG Schema for MathML3, discusses the RelaxNG Schema for MathML3 [RELAX-NG], which forms a normative part of the specification. Following that, Section A.4 Using the MathML XML Schema, and Section A.3 Using the MathML DTD discuss alternative languages definition using the document type definitions (DTD) and the W3C XML schema language, [XMLSchemas], both of which are derived from the normative RelaxNG schema automatically. One should note that the schema definitions of the language is currently stricter than the DTD version. That is, a schema validating processor will declare invalid documents that are declared valid by a (DTD) validating XML parser. This is partly due to the fact that the XML schema language may express additional constraints not expressable in the DTD, and partly due to the fact that for reasons of compatibility with earlier releases, the DTD is intentionally forgiving in some places and does not enforce constraints that are specified in the text of this specification.
MathML documents should be validated using the RelaxNG Schema for MathML, either in the XML encoding (http://www.w3.org/Math/RelaxNG/mathml3/mathml3.rng) or in compact notation (http://www.w3.org/Math/RelaxNG/mathml3/mathml3.rnc) which is also shown below.
In contrast to DTDs there is no in-document method to associate a RelaxNG schema with a document.
We provide five RelaxNG schema for MathML3 in five parts:
The grammar for full MathML
The grammar for elements common to Content and Presentation
The grammar for Presentation MathML
The grammar for Strict Content MathML
The grammar for Content MathML3
The RelaxNG schema for full MathML builds on the schema describing the various parts of the language which are given in the following sections. It can be found at http://www.w3.org/Math/RelaxNG/mathml3/mathml3.rnc.
# This is the Mathematical Markup Language (MathML) 3.0, an XML # application for describing mathematical notation and capturing # both its structure and content. # # Copyright 1998-2010 W3C (MIT, ERCIM, Keio) # # Use and distribution of this code are permitted under the terms # W3C Software Notice and License # http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231 default namespace m = "http://www.w3.org/1998/Math/MathML" ## Content MathML include "mathml3-content.rnc" ## Presentation MathML include "mathml3-presentation.rnc" ## math and semantics common to both Content and Presentation include "mathml3-common.rnc"
# This is the Mathematical Markup Language (MathML) 3.0, an XML # application for describing mathematical notation and capturing # both its structure and content. # # Copyright 1998-2014 W3C (MIT, ERCIM, Keio, Beihang) # # Use and distribution of this code are permitted under the terms # W3C Software Notice and License # http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231 default namespace m = "http://www.w3.org/1998/Math/MathML" namespace local = "" start = math math = elementmath
{math.attributes,MathExpression*} MathExpression = semantics NonMathMLAtt = attribute(* - (local:*|m:*)
) {xsd:string} CommonDeprecatedAtt = attributeother
{text}? CommonAtt = attributeid
{xsd:ID}?, attributexref
{text}?, attributeclass
{xsd:NMTOKENS}?, attributestyle
{xsd:string}?, attributehref
{xsd:anyURI}?, CommonDeprecatedAtt, NonMathMLAtt* math.attributes = CommonAtt, attributedisplay
{"block" | "inline"}?, attributemaxwidth
{length}?, attributeoverflow
{"linebreak" | "scroll" | "elide" | "truncate" | "scale"}?, attributealtimg
{xsd:anyURI}?, attributealtimg-width
{length}?, attributealtimg-height
{length}?, attributealtimg-valign
{length | "top" | "middle" | "bottom"}?, attributealttext
{text}?, attributecdgroup
{xsd:anyURI}?, math.deprecatedattributes # the mathml3-presentation schema adds additional attributes # to the math element, all those valid on mstyle math.deprecatedattributes = attributemode
{xsd:string}?, attributemacros
{xsd:string}? name = attributename
{xsd:NCName} cd = attributecd
{xsd:NCName} src = attributesrc
{xsd:anyURI}? annotation = elementannotation
{annotation.attributes,text} annotation-xml.model = (MathExpression|anyElement)* anyElement = element (* - m:*) {(attribute*
{text}|text| anyElement)*} annotation-xml = elementannotation-xml
{annotation.attributes, annotation-xml.model} annotation.attributes = CommonAtt, cd?, name?, DefEncAtt, src? DefEncAtt = attributeencoding
{xsd:string}?, attributedefinitionURL
{xsd:anyURI}? semantics = elementsemantics
{semantics.attributes, MathExpression, (annotation|annotation-xml)*} semantics.attributes = CommonAtt,DefEncAtt,cd?,name? length # wrapped for display = xsd:string { pattern = '\s*((-?[0-9]*([0-9]\.?|\.[0-9])[0-9]*(e[mx]|in|cm|mm|p[xtc]|%)?)|(negative)?((very){0,2}thi(n| ck)|medium)mathspace)\s*' }
# This is the Mathematical Markup Language (MathML) 3.0, an XML # application for describing mathematical notation and capturing # both its structure and content. # # Copyright 1998-2014 W3C (MIT, ERCIM, Keio, Beihang) # # Use and distribution of this code are permitted under the terms # W3C Software Notice and License # http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231 default namespace m = "http://www.w3.org/1998/Math/MathML" MathExpression |= PresentationExpression ImpliedMrow = MathExpression* TableRowExpression = mtr|mlabeledtr TableCellExpression = mtd MstackExpression = MathExpression|mscarries|msline|msrow|msgroup MsrowExpression = MathExpression|none MultiScriptExpression = (MathExpression|none),(MathExpression|none) mpadded-length # wrapped for display = xsd:string { pattern = '\s*([\+\-]?[0-9]*([0-9]\.?|\.[0-9])[0-9]*\s*((%?\s*(height|depth|width)?)| e[mx]|in|cm|mm|p[xtc]|((negative)?((very){0,2}thi(n|ck)|medium)mathspace))?)\s*' } linestyle = "none" | "solid" | "dashed" verticalalign = "top" | "bottom" | "center" | "baseline" | "axis" columnalignstyle = "left" | "center" | "right" notationstyle = "longdiv" | "actuarial" | "radical" | "box" | "roundedbox" | "circle" | "left" | "right" | "top" | "bottom" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike" | "madruwb" idref = text unsigned-integer = xsd:unsignedLong integer = xsd:integer number = xsd:decimal character = xsd:string { pattern = '\s*\S\s*'} color # wrapped for display = xsd:string { pattern = '\s*((#[0-9a-fA-F]{3}([0-9a-fA-F]{3})?)|[aA][qQ][uU][aA]|[bB][lL][aA][cC][kK]| [bB][lL][uU][eE]|[fF][uU][cC][hH][sS][iI][aA]|[gG][rR][aA][yY]|[gG][rR][eE][eE][nN]| [lL][iI][mM][eE]|[mM][aA][rR][oO][oO][nN]|[nN][aA][vV][yY]|[oO][lL][iI][vV][eE]|[pP][uU][rR][pP][lL][eE]| [rR][eE][dD]|[sS][iI][lL][vV][eE][rR]|[tT][eE][aA][lL]|[wW][hH][iI][tT][eE]|[yY][eE][lL][lL][oO][wW])\s*'} group-alignment = "left" | "center" | "right" | "decimalpoint" group-alignment-list = list {group-alignment+} group-alignment-list-list = xsd:string { pattern = '(\s*\{\s*(left|center|right|decimalpoint)(\s+(left|center|right|decimalpoint))*\})*\s*' } positive-integer = xsd:positiveInteger TokenExpression = mi|mn|mo|mtext|mspace|ms token.content = mglyph|malignmark|text mi = elementmi
{mi.attributes, token.content*} mi.attributes = CommonAtt, CommonPresAtt, TokenAtt mn = elementmn
{mn.attributes, token.content*} mn.attributes = CommonAtt, CommonPresAtt, TokenAtt mo = elementmo
{mo.attributes, token.content*} mo.attributes = CommonAtt, CommonPresAtt, TokenAtt, attributeform
{"prefix" | "infix" | "postfix"}?, attributefence
{"true" | "false"}?, attributeseparator
{"true" | "false"}?, attributelspace
{length}?, attributerspace
{length}?, attributestretchy
{"true" | "false"}?, attributesymmetric
{"true" | "false"}?, attributemaxsize
{length | "infinity"}?, attributeminsize
{length}?, attributelargeop
{"true" | "false"}?, attributemovablelimits
{"true" | "false"}?, attributeaccent
{"true" | "false"}?, attributelinebreak
{"auto" | "newline" | "nobreak" | "goodbreak" | "badbreak"}?, attributelineleading
{length}?, attributelinebreakstyle
{"before" | "after" | "duplicate" | "infixlinebreakstyle"}?, attributelinebreakmultchar
{text}?, attributeindentalign
{"left" | "center" | "right" | "auto" | "id"}?, attributeindentshift
{length}?, attributeindenttarget
{idref}?, attributeindentalignfirst
{"left" | "center" | "right" | "auto" | "id" | "indentalign"}?, attributeindentshiftfirst
{length | "indentshift"}?, attributeindentalignlast
{"left" | "center" | "right" | "auto" | "id" | "indentalign"}?, attributeindentshiftlast
{length | "indentshift"}? mtext = elementmtext
{mtext.attributes, token.content*} mtext.attributes = CommonAtt, CommonPresAtt, TokenAtt mspace = elementmspace
{mspace.attributes, empty} mspace.attributes = CommonAtt, CommonPresAtt, TokenAtt, attributewidth
{length}?, attributeheight
{length}?, attributedepth
{length}?, attributelinebreak
{"auto" | "newline" | "nobreak" | "goodbreak" | "badbreak" | "indentingnewline"}?, attributeindentalign
{"left" | "center" | "right" | "auto" | "id"}?, attributeindentshift
{length}?, attributeindenttarget
{idref}?, attributeindentalignfirst
{"left" | "center" | "right" | "auto" | "id" | "indentalign"}?, attributeindentshiftfirst
{length | "indentshift"}?, attributeindentalignlast
{"left" | "center" | "right" | "auto" | "id" | "indentalign"}?, attributeindentshiftlast
{length | "indentshift"}? ms = elementms
{ms.attributes, token.content*} ms.attributes = CommonAtt, CommonPresAtt, TokenAtt, attributelquote
{text}?, attributerquote
{text}? mglyph = elementmglyph
{mglyph.attributes,mglyph.deprecatedattributes,empty} mglyph.attributes = CommonAtt, CommonPresAtt, attributesrc
{xsd:anyURI}?, attributewidth
{length}?, attributeheight
{length}?, attributevalign
{length}?, attributealt
{text}? mglyph.deprecatedattributes = attributeindex
{integer}?, attributemathvariant
{"normal" | "bold" | "italic" | "bold-italic" | "double-struck" | "bold-fraktur" | "script" | "bold-script" | "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" | "sans-serif-bold-italic" | "monospace" | "initial" | "tailed" | "looped" | "stretched"}?, attributemathsize
{"small" | "normal" | "big" | length}?, DeprecatedTokenAtt msline = elementmsline
{msline.attributes,empty} msline.attributes = CommonAtt, CommonPresAtt, attributeposition
{integer}?, attributelength
{unsigned-integer}?, attributeleftoverhang
{length}?, attributerightoverhang
{length}?, attributemslinethickness
{length | "thin" | "medium" | "thick"}? none = elementnone
{none.attributes,empty} none.attributes = CommonAtt, CommonPresAtt mprescripts = elementmprescripts
{mprescripts.attributes,empty} mprescripts.attributes = CommonAtt, CommonPresAtt CommonPresAtt = attributemathcolor
{color}?, attributemathbackground
{color | "transparent"}? TokenAtt = attributemathvariant
{"normal" | "bold" | "italic" | "bold-italic" | "double-struck" | "bold-fraktur" | "script" | "bold-script" | "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" | "sans-serif-bold-italic" | "monospace" | "initial" | "tailed" | "looped" | "stretched"}?, attributemathsize
{"small" | "normal" | "big" | length}?, attributedir
{"ltr" | "rtl"}?, DeprecatedTokenAtt DeprecatedTokenAtt = attributefontfamily
{text}?, attributefontweight
{"normal" | "bold"}?, attributefontstyle
{"normal" | "italic"}?, attributefontsize
{length}?, attributecolor
{color}?, attributebackground
{color | "transparent"}? MalignExpression = maligngroup|malignmark malignmark = elementmalignmark
{malignmark.attributes, empty} malignmark.attributes = CommonAtt, CommonPresAtt, attributeedge
{"left" | "right"}? maligngroup = elementmaligngroup
{maligngroup.attributes, empty} maligngroup.attributes = CommonAtt, CommonPresAtt, attributegroupalign
{"left" | "center" | "right" | "decimalpoint"}? PresentationExpression = TokenExpression|MalignExpression| mrow|mfrac|msqrt|mroot|mstyle|merror|mpadded|mphantom| mfenced|menclose|msub|msup|msubsup|munder|mover|munderover| mmultiscripts|mtable|mstack|mlongdiv|maction mrow = elementmrow
{mrow.attributes, MathExpression*} mrow.attributes = CommonAtt, CommonPresAtt, attributedir
{"ltr" | "rtl"}? mfrac = elementmfrac
{mfrac.attributes, MathExpression, MathExpression} mfrac.attributes = CommonAtt, CommonPresAtt, attributelinethickness
{length | "thin" | "medium" | "thick"}?, attributenumalign
{"left" | "center" | "right"}?, attributedenomalign
{"left" | "center" | "right"}?, attributebevelled
{"true" | "false"}? msqrt = elementmsqrt
{msqrt.attributes, ImpliedMrow} msqrt.attributes = CommonAtt, CommonPresAtt mroot = elementmroot
{mroot.attributes, MathExpression, MathExpression} mroot.attributes = CommonAtt, CommonPresAtt mstyle = elementmstyle
{mstyle.attributes, ImpliedMrow} mstyle.attributes = CommonAtt, CommonPresAtt, mstyle.specificattributes, mstyle.generalattributes, mstyle.deprecatedattributes mstyle.specificattributes = attributescriptlevel
{integer}?, attributedisplaystyle
{"true" | "false"}?, attributescriptsizemultiplier
{number}?, attributescriptminsize
{length}?, attributeinfixlinebreakstyle
{"before" | "after" | "duplicate"}?, attributedecimalpoint
{character}? mstyle.generalattributes = attributeaccent
{"true" | "false"}?, attributeaccentunder
{"true" | "false"}?, attributealign
{"left" | "right" | "center"}?, attributealignmentscope
{list {("true" | "false") +}}?, attributebevelled
{"true" | "false"}?, attributecharalign
{"left" | "center" | "right"}?, attributecharspacing
{length | "loose" | "medium" | "tight"}?, attributeclose
{text}?, attributecolumnalign
{list {columnalignstyle+} }?, attributecolumnlines
{list {linestyle +}}?, attributecolumnspacing
{list {(length) +}}?, attributecolumnspan
{positive-integer}?, attributecolumnwidth
{list {("auto" | length | "fit") +}}?, attributecrossout
{list {("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")*}}?, attributedenomalign
{"left" | "center" | "right"}?, attributedepth
{length}?, attributedir
{"ltr" | "rtl"}?, attributeedge
{"left" | "right"}?, attributeequalcolumns
{"true" | "false"}?, attributeequalrows
{"true" | "false"}?, attributefence
{"true" | "false"}?, attributeform
{"prefix" | "infix" | "postfix"}?, attributeframe
{linestyle}?, attributeframespacing
{list {length, length}}?, attributegroupalign
{group-alignment-list-list}?, attributeheight
{length}?, attributeindentalign
{"left" | "center" | "right" | "auto" | "id"}?, attributeindentalignfirst
{"left" | "center" | "right" | "auto" | "id" | "indentalign"}?, attributeindentalignlast
{"left" | "center" | "right" | "auto" | "id" | "indentalign"}?, attributeindentshift
{length}?, attributeindentshiftfirst
{length | "indentshift"}?, attributeindentshiftlast
{length | "indentshift"}?, attributeindenttarget
{idref}?, attributelargeop
{"true" | "false"}?, attributeleftoverhang
{length}?, attributelength
{unsigned-integer}?, attributelinebreak
{"auto" | "newline" | "nobreak" | "goodbreak" | "badbreak"}?, attributelinebreakmultchar
{text}?, attributelinebreakstyle
{"before" | "after" | "duplicate" | "infixlinebreakstyle"}?, attributelineleading
{length}?, attributelinethickness
{length | "thin" | "medium" | "thick"}?, attributelocation
{"w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw"}?, attributelongdivstyle
{"lefttop" | "stackedrightright" | "mediumstackedrightright" | "shortstackedrightright" | "righttop" | "left/\right" | "left)(right" | ":right=right" | "stackedleftleft" | "stackedleftlinetop"}?, attributelquote
{text}?, attributelspace
{length}?, attributemathsize
{"small" | "normal" | "big" | length}?, attributemathvariant
{"normal" | "bold" | "italic" | "bold-italic" | "double-struck" | "bold-fraktur" | "script" | "bold-script" | "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" | "sans-serif-bold-italic" | "monospace" | "initial" | "tailed" | "looped" | "stretched"}?, attributemaxsize
{length | "infinity"}?, attributeminlabelspacing
{length}?, attributeminsize
{length}?, attributemovablelimits
{"true" | "false"}?, attributemslinethickness
{length | "thin" | "medium" | "thick"}?, attributenotation
{text}?, attributenumalign
{"left" | "center" | "right"}?, attributeopen
{text}?, attributeposition
{integer}?, attributerightoverhang
{length}?, attributerowalign
{list {verticalalign+} }?, attributerowlines
{list {linestyle +}}?, attributerowspacing
{list {(length) +}}?, attributerowspan
{positive-integer}?, attributerquote
{text}?, attributerspace
{length}?, attributeselection
{positive-integer}?, attributeseparator
{"true" | "false"}?, attributeseparators
{text}?, attributeshift
{integer}?, attributeside
{"left" | "right" | "leftoverlap" | "rightoverlap"}?, attributestackalign
{"left" | "center" | "right" | "decimalpoint"}?, attributestretchy
{"true" | "false"}?, attributesubscriptshift
{length}?, attributesuperscriptshift
{length}?, attributesymmetric
{"true" | "false"}?, attributevalign
{length}?, attributewidth
{length}? mstyle.deprecatedattributes = DeprecatedTokenAtt, attributeveryverythinmathspace
{length}?, attributeverythinmathspace
{length}?, attributethinmathspace
{length}?, attributemediummathspace
{length}?, attributethickmathspace
{length}?, attributeverythickmathspace
{length}?, attributeveryverythickmathspace
{length}? math.attributes &= CommonPresAtt math.attributes &= mstyle.specificattributes math.attributes &= mstyle.generalattributes merror = elementmerror
{merror.attributes, ImpliedMrow} merror.attributes = CommonAtt, CommonPresAtt mpadded = elementmpadded
{mpadded.attributes, ImpliedMrow} mpadded.attributes = CommonAtt, CommonPresAtt, attributeheight
{mpadded-length}?, attributedepth
{mpadded-length}?, attributewidth
{mpadded-length}?, attributelspace
{mpadded-length}?, attributevoffset
{mpadded-length}? mphantom = elementmphantom
{mphantom.attributes, ImpliedMrow} mphantom.attributes = CommonAtt, CommonPresAtt mfenced = elementmfenced
{mfenced.attributes, MathExpression*} mfenced.attributes = CommonAtt, CommonPresAtt, attributeopen
{text}?, attributeclose
{text}?, attributeseparators
{text}? menclose = elementmenclose
{menclose.attributes, ImpliedMrow} menclose.attributes = CommonAtt, CommonPresAtt, attributenotation
{text}? msub = elementmsub
{msub.attributes, MathExpression, MathExpression} msub.attributes = CommonAtt, CommonPresAtt, attributesubscriptshift
{length}? msup = elementmsup
{msup.attributes, MathExpression, MathExpression} msup.attributes = CommonAtt, CommonPresAtt, attributesuperscriptshift
{length}? msubsup = elementmsubsup
{msubsup.attributes, MathExpression, MathExpression, MathExpression} msubsup.attributes = CommonAtt, CommonPresAtt, attributesubscriptshift
{length}?, attributesuperscriptshift
{length}? munder = elementmunder
{munder.attributes, MathExpression, MathExpression} munder.attributes = CommonAtt, CommonPresAtt, attributeaccentunder
{"true" | "false"}?, attributealign
{"left" | "right" | "center"}? mover = elementmover
{mover.attributes, MathExpression, MathExpression} mover.attributes = CommonAtt, CommonPresAtt, attributeaccent
{"true" | "false"}?, attributealign
{"left" | "right" | "center"}? munderover = elementmunderover
{munderover.attributes, MathExpression, MathExpression, MathExpression} munderover.attributes = CommonAtt, CommonPresAtt, attributeaccent
{"true" | "false"}?, attributeaccentunder
{"true" | "false"}?, attributealign
{"left" | "right" | "center"}? mmultiscripts = elementmmultiscripts
{mmultiscripts.attributes, MathExpression,MultiScriptExpression*,(mprescripts,MultiScriptExpression*)?} mmultiscripts.attributes = msubsup.attributes mtable = elementmtable
{mtable.attributes, TableRowExpression*} mtable.attributes = CommonAtt, CommonPresAtt, attributealign
{xsd:string { pattern ='\s*(top|bottom|center|baseline|axis)(\s+-?[0-9]+)?\s*'}}?, attributerowalign
{list {verticalalign+} }?, attributecolumnalign
{list {columnalignstyle+} }?, attributegroupalign
{group-alignment-list-list}?, attributealignmentscope
{list {("true" | "false") +}}?, attributecolumnwidth
{list {("auto" | length | "fit") +}}?, attributewidth
{"auto" | length}?, attributerowspacing
{list {(length) +}}?, attributecolumnspacing
{list {(length) +}}?, attributerowlines
{list {linestyle +}}?, attributecolumnlines
{list {linestyle +}}?, attributeframe
{linestyle}?, attributeframespacing
{list {length, length}}?, attributeequalrows
{"true" | "false"}?, attributeequalcolumns
{"true" | "false"}?, attributedisplaystyle
{"true" | "false"}?, attributeside
{"left" | "right" | "leftoverlap" | "rightoverlap"}?, attributeminlabelspacing
{length}? mlabeledtr = elementmlabeledtr
{mlabeledtr.attributes, TableCellExpression+} mlabeledtr.attributes = mtr.attributes mtr = elementmtr
{mtr.attributes, TableCellExpression*} mtr.attributes = CommonAtt, CommonPresAtt, attributerowalign
{"top" | "bottom" | "center" | "baseline" | "axis"}?, attributecolumnalign
{list {columnalignstyle+} }?, attributegroupalign
{group-alignment-list-list}? mtd = elementmtd
{mtd.attributes, ImpliedMrow} mtd.attributes = CommonAtt, CommonPresAtt, attributerowspan
{positive-integer}?, attributecolumnspan
{positive-integer}?, attributerowalign
{"top" | "bottom" | "center" | "baseline" | "axis"}?, attributecolumnalign
{columnalignstyle}?, attributegroupalign
{group-alignment-list}? mstack = elementmstack
{mstack.attributes, MstackExpression*} mstack.attributes = CommonAtt, CommonPresAtt, attributealign
{xsd:string { pattern ='\s*(top|bottom|center|baseline|axis)(\s+-?[0-9]+)?\s*'}}?, attributestackalign
{"left" | "center" | "right" | "decimalpoint"}?, attributecharalign
{"left" | "center" | "right"}?, attributecharspacing
{length | "loose" | "medium" | "tight"}? mlongdiv = elementmlongdiv
{mlongdiv.attributes, MstackExpression,MstackExpression,MstackExpression+} mlongdiv.attributes = msgroup.attributes, attributelongdivstyle
{"lefttop" | "stackedrightright" | "mediumstackedrightright" | "shortstackedrightright" | "righttop" | "left/\right" | "left)(right" | ":right=right" | "stackedleftleft" | "stackedleftlinetop"}? msgroup = elementmsgroup
{msgroup.attributes, MstackExpression*} msgroup.attributes = CommonAtt, CommonPresAtt, attributeposition
{integer}?, attributeshift
{integer}? msrow = elementmsrow
{msrow.attributes, MsrowExpression*} msrow.attributes = CommonAtt, CommonPresAtt, attributeposition
{integer}? mscarries = elementmscarries
{mscarries.attributes, (MsrowExpression|mscarry)*} mscarries.attributes = CommonAtt, CommonPresAtt, attributeposition
{integer}?, attributelocation
{"w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw"}?, attributecrossout
{list {("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")*}}?, attributescriptsizemultiplier
{number}? mscarry = elementmscarry
{mscarry.attributes, MsrowExpression*} mscarry.attributes = CommonAtt, CommonPresAtt, attributelocation
{"w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw"}?, attributecrossout
{list {("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")*}}? maction = elementmaction
{maction.attributes, MathExpression+} maction.attributes = CommonAtt, CommonPresAtt, attributeactiontype
{text}, attributeselection
{positive-integer}?
The grammar for Strict Content MathML3 can be found at http://www.w3.org/Math/RelaxNG/mathml3/mathml3-strict-content.rnc.
# This is the Mathematical Markup Language (MathML) 3.0, an XML # application for describing mathematical notation and capturing # both its structure and content. # # Copyright 1998-2014 W3C (MIT, ERCIM, Keio, Beihang) # # Use and distribution of this code are permitted under the terms # W3C Software Notice and License # http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231 default namespace m = "http://www.w3.org/1998/Math/MathML" ContExp = semantics-contexp | cn | ci | csymbol | apply | bind | share | cerror | cbytes | cs cn = elementcn
{cn.attributes,cn.content} cn.content = text cn.attributes = CommonAtt, attributetype
{"integer" | "real" | "double" | "hexdouble"} semantics-ci = elementsemantics
{semantics.attributes,(ci|semantics-ci), (annotation|annotation-xml)*} semantics-contexp = elementsemantics
{semantics.attributes,ContExp, (annotation|annotation-xml)*} ci = elementci
{ci.attributes, ci.content} ci.attributes = CommonAtt, ci.type? ci.type = attributetype
{"integer" | "rational" | "real" | "complex" | "complex-polar" | "complex-cartesian" | "constant" | "function" | "vector" | "list" | "set" | "matrix"} ci.content = text csymbol = elementcsymbol
{csymbol.attributes,csymbol.content} SymbolName = xsd:NCName csymbol.attributes = CommonAtt, cd csymbol.content = SymbolName BvarQ = bvar* bvar = elementbvar
{CommonAtt, (ci | semantics-ci)} apply = elementapply
{CommonAtt,apply.content} apply.content = ContExp+ bind = elementbind
{CommonAtt,bind.content} bind.content = ContExp,bvar*,ContExp share = elementshare
{CommonAtt, src, empty} cerror = elementcerror
{cerror.attributes, csymbol, ContExp*} cerror.attributes = CommonAtt cbytes = elementcbytes
{cbytes.attributes, base64} cbytes.attributes = CommonAtt base64 = xsd:base64Binary cs = elementcs
{cs.attributes, text} cs.attributes = CommonAtt MathExpression |= ContExp
The grammar for Content MathML3 builds on the grammar for the Strict Content MathML subset, and can be found at http://www.w3.org/Math/RelaxNG/mathml3/mathml3-content.rnc.
# This is the Mathematical Markup Language (MathML) 3.0, an XML # application for describing mathematical notation and capturing # both its structure and content. # # Copyright 1998-2014 W3C (MIT, ERCIM, Keio, Beihang) # # Use and distribution of this code are permitted under the terms # W3C Software Notice and License # http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231 include "mathml3-strict-content.rnc"{ cn.content = (text | mglyph | sep | PresentationExpression)* cn.attributes = CommonAtt, DefEncAtt, attributetype
{text}?, base? ci.attributes = CommonAtt, DefEncAtt, ci.type? ci.type = attributetype
{text} ci.content = (text | mglyph | PresentationExpression)* csymbol.attributes = CommonAtt, DefEncAtt, attributetype
{text}?,cd? csymbol.content = (text | mglyph | PresentationExpression)* bvar = elementbvar
{CommonAtt, ((ci | semantics-ci) & degree?)} cbytes.attributes = CommonAtt, DefEncAtt cs.attributes = CommonAtt, DefEncAtt apply.content = ContExp+ | (ContExp, BvarQ, Qualifier*, ContExp*) bind.content = apply.content } base = attributebase
{text} sep = elementsep
{empty} PresentationExpression |= notAllowed DomainQ = (domainofapplication|condition|interval|(lowlimit,uplimit?))* domainofapplication = elementdomainofapplication
{ContExp} condition = elementcondition
{ContExp} uplimit = elementuplimit
{ContExp} lowlimit = elementlowlimit
{ContExp} Qualifier = DomainQ|degree|momentabout|logbase degree = elementdegree
{ContExp} momentabout = elementmomentabout
{ContExp} logbase = elementlogbase
{ContExp} type = attributetype
{text} order = attributeorder
{"numeric" | "lexicographic"} closure = attributeclosure
{text} ContExp |= piecewise piecewise = elementpiecewise
{CommonAtt, DefEncAtt,(piece* & otherwise?)} piece = elementpiece
{CommonAtt, DefEncAtt, ContExp, ContExp} otherwise = elementotherwise
{CommonAtt, DefEncAtt, ContExp} DeprecatedContExp = reln | fn | declare ContExp |= DeprecatedContExp reln = elementreln
{ContExp*} fn = elementfn
{ContExp} declare = elementdeclare
{attributetype
{xsd:string}?, attributescope
{xsd:string}?, attributenargs
{xsd:nonNegativeInteger}?, attributeoccurrence
{"prefix"|"infix"|"function-model"}?, DefEncAtt, ContExp+} interval.class = interval ContExp |= interval.class interval = elementinterval
{ CommonAtt, DefEncAtt,closure?, ContExp,ContExp} unary-functional.class = inverse | ident | domain | codomain | image | ln | log | moment ContExp |= unary-functional.class inverse = elementinverse
{ CommonAtt, DefEncAtt, empty} ident = elementident
{ CommonAtt, DefEncAtt, empty} domain = elementdomain
{ CommonAtt, DefEncAtt, empty} codomain = elementcodomain
{ CommonAtt, DefEncAtt, empty} image = elementimage
{ CommonAtt, DefEncAtt, empty} ln = elementln
{ CommonAtt, DefEncAtt, empty} log = elementlog
{ CommonAtt, DefEncAtt, empty} moment = elementmoment
{ CommonAtt, DefEncAtt, empty} lambda.class = lambda ContExp |= lambda.class lambda = elementlambda
{ CommonAtt, DefEncAtt, BvarQ, DomainQ, ContExp} nary-functional.class = compose ContExp |= nary-functional.class compose = elementcompose
{ CommonAtt, DefEncAtt, empty} binary-arith.class = quotient | divide | minus | power | rem | root ContExp |= binary-arith.class quotient = elementquotient
{ CommonAtt, DefEncAtt, empty} divide = elementdivide
{ CommonAtt, DefEncAtt, empty} minus = elementminus
{ CommonAtt, DefEncAtt, empty} power = elementpower
{ CommonAtt, DefEncAtt, empty} rem = elementrem
{ CommonAtt, DefEncAtt, empty} root = elementroot
{ CommonAtt, DefEncAtt, empty} unary-arith.class = factorial | minus | root | abs | conjugate | arg | real | imaginary | floor | ceiling | exp ContExp |= unary-arith.class factorial = elementfactorial
{ CommonAtt, DefEncAtt, empty} abs = elementabs
{ CommonAtt, DefEncAtt, empty} conjugate = elementconjugate
{ CommonAtt, DefEncAtt, empty} arg = elementarg
{ CommonAtt, DefEncAtt, empty} real = elementreal
{ CommonAtt, DefEncAtt, empty} imaginary = elementimaginary
{ CommonAtt, DefEncAtt, empty} floor = elementfloor
{ CommonAtt, DefEncAtt, empty} ceiling = elementceiling
{ CommonAtt, DefEncAtt, empty} exp = elementexp
{ CommonAtt, DefEncAtt, empty} nary-minmax.class = max | min ContExp |= nary-minmax.class max = elementmax
{ CommonAtt, DefEncAtt, empty} min = elementmin
{ CommonAtt, DefEncAtt, empty} nary-arith.class = plus | times | gcd | lcm ContExp |= nary-arith.class plus = elementplus
{ CommonAtt, DefEncAtt, empty} times = elementtimes
{ CommonAtt, DefEncAtt, empty} gcd = elementgcd
{ CommonAtt, DefEncAtt, empty} lcm = elementlcm
{ CommonAtt, DefEncAtt, empty} nary-logical.class = and | or | xor ContExp |= nary-logical.class and = elementand
{ CommonAtt, DefEncAtt, empty} or = elementor
{ CommonAtt, DefEncAtt, empty} xor = elementxor
{ CommonAtt, DefEncAtt, empty} unary-logical.class = not ContExp |= unary-logical.class not = elementnot
{ CommonAtt, DefEncAtt, empty} binary-logical.class = implies | equivalent ContExp |= binary-logical.class implies = elementimplies
{ CommonAtt, DefEncAtt, empty} equivalent = elementequivalent
{ CommonAtt, DefEncAtt, empty} quantifier.class = forall | exists ContExp |= quantifier.class forall = elementforall
{ CommonAtt, DefEncAtt, empty} exists = elementexists
{ CommonAtt, DefEncAtt, empty} nary-reln.class = eq | gt | lt | geq | leq ContExp |= nary-reln.class eq = elementeq
{ CommonAtt, DefEncAtt, empty} gt = elementgt
{ CommonAtt, DefEncAtt, empty} lt = elementlt
{ CommonAtt, DefEncAtt, empty} geq = elementgeq
{ CommonAtt, DefEncAtt, empty} leq = elementleq
{ CommonAtt, DefEncAtt, empty} binary-reln.class = neq | approx | factorof | tendsto ContExp |= binary-reln.class neq = elementneq
{ CommonAtt, DefEncAtt, empty} approx = elementapprox
{ CommonAtt, DefEncAtt, empty} factorof = elementfactorof
{ CommonAtt, DefEncAtt, empty} tendsto = elementtendsto
{ CommonAtt, DefEncAtt, type?, empty} int.class = int ContExp |= int.class int = elementint
{ CommonAtt, DefEncAtt, empty} Differential-Operator.class = diff ContExp |= Differential-Operator.class diff = elementdiff
{ CommonAtt, DefEncAtt, empty} partialdiff.class = partialdiff ContExp |= partialdiff.class partialdiff = elementpartialdiff
{ CommonAtt, DefEncAtt, empty} unary-veccalc.class = divergence | grad | curl | laplacian ContExp |= unary-veccalc.class divergence = elementdivergence
{ CommonAtt, DefEncAtt, empty} grad = elementgrad
{ CommonAtt, DefEncAtt, empty} curl = elementcurl
{ CommonAtt, DefEncAtt, empty} laplacian = elementlaplacian
{ CommonAtt, DefEncAtt, empty} nary-setlist-constructor.class = set | \list ContExp |= nary-setlist-constructor.class set = elementset
{ CommonAtt, DefEncAtt, type?, BvarQ*, DomainQ*, ContExp*} \list = element\list
{ CommonAtt, DefEncAtt, order?, BvarQ*, DomainQ*, ContExp*} nary-set.class = union | intersect | cartesianproduct ContExp |= nary-set.class union = elementunion
{ CommonAtt, DefEncAtt, empty} intersect = elementintersect
{ CommonAtt, DefEncAtt, empty} cartesianproduct = elementcartesianproduct
{ CommonAtt, DefEncAtt, empty} binary-set.class = in | notin | notsubset | notprsubset | setdiff ContExp |= binary-set.class in = elementin
{ CommonAtt, DefEncAtt, empty} notin = elementnotin
{ CommonAtt, DefEncAtt, empty} notsubset = elementnotsubset
{ CommonAtt, DefEncAtt, empty} notprsubset = elementnotprsubset
{ CommonAtt, DefEncAtt, empty} setdiff = elementsetdiff
{ CommonAtt, DefEncAtt, empty} nary-set-reln.class = subset | prsubset ContExp |= nary-set-reln.class subset = elementsubset
{ CommonAtt, DefEncAtt, empty} prsubset = elementprsubset
{ CommonAtt, DefEncAtt, empty} unary-set.class = card ContExp |= unary-set.class card = elementcard
{ CommonAtt, DefEncAtt, empty} sum.class = sum ContExp |= sum.class sum = elementsum
{ CommonAtt, DefEncAtt, empty} product.class = product ContExp |= product.class product = elementproduct
{ CommonAtt, DefEncAtt, empty} limit.class = limit ContExp |= limit.class limit = elementlimit
{ CommonAtt, DefEncAtt, empty} unary-elementary.class = sin | cos | tan | sec | csc | cot | sinh | cosh | tanh | sech | csch | coth | arcsin | arccos | arctan | arccosh | arccot | arccoth | arccsc | arccsch | arcsec | arcsech | arcsinh | arctanh ContExp |= unary-elementary.class sin = elementsin
{ CommonAtt, DefEncAtt, empty} cos = elementcos
{ CommonAtt, DefEncAtt, empty} tan = elementtan
{ CommonAtt, DefEncAtt, empty} sec = elementsec
{ CommonAtt, DefEncAtt, empty} csc = elementcsc
{ CommonAtt, DefEncAtt, empty} cot = elementcot
{ CommonAtt, DefEncAtt, empty} sinh = elementsinh
{ CommonAtt, DefEncAtt, empty} cosh = elementcosh
{ CommonAtt, DefEncAtt, empty} tanh = elementtanh
{ CommonAtt, DefEncAtt, empty} sech = elementsech
{ CommonAtt, DefEncAtt, empty} csch = elementcsch
{ CommonAtt, DefEncAtt, empty} coth = elementcoth
{ CommonAtt, DefEncAtt, empty} arcsin = elementarcsin
{ CommonAtt, DefEncAtt, empty} arccos = elementarccos
{ CommonAtt, DefEncAtt, empty} arctan = elementarctan
{ CommonAtt, DefEncAtt, empty} arccosh = elementarccosh
{ CommonAtt, DefEncAtt, empty} arccot = elementarccot
{ CommonAtt, DefEncAtt, empty} arccoth = elementarccoth
{ CommonAtt, DefEncAtt, empty} arccsc = elementarccsc
{ CommonAtt, DefEncAtt, empty} arccsch = elementarccsch
{ CommonAtt, DefEncAtt, empty} arcsec = elementarcsec
{ CommonAtt, DefEncAtt, empty} arcsech = elementarcsech
{ CommonAtt, DefEncAtt, empty} arcsinh = elementarcsinh
{ CommonAtt, DefEncAtt, empty} arctanh = elementarctanh
{ CommonAtt, DefEncAtt, empty} nary-stats.class = mean | sdev | variance | median | mode ContExp |= nary-stats.class mean = elementmean
{ CommonAtt, DefEncAtt, empty} sdev = elementsdev
{ CommonAtt, DefEncAtt, empty} variance = elementvariance
{ CommonAtt, DefEncAtt, empty} median = elementmedian
{ CommonAtt, DefEncAtt, empty} mode = elementmode
{ CommonAtt, DefEncAtt, empty} nary-constructor.class = vector | matrix | matrixrow ContExp |= nary-constructor.class vector = elementvector
{ CommonAtt, DefEncAtt, BvarQ, DomainQ, ContExp*} matrix = elementmatrix
{ CommonAtt, DefEncAtt, BvarQ, DomainQ, ContExp*} matrixrow = elementmatrixrow
{ CommonAtt, DefEncAtt, BvarQ, DomainQ, ContExp*} unary-linalg.class = determinant | transpose ContExp |= unary-linalg.class determinant = elementdeterminant
{ CommonAtt, DefEncAtt, empty} transpose = elementtranspose
{ CommonAtt, DefEncAtt, empty} nary-linalg.class = selector ContExp |= nary-linalg.class selector = elementselector
{ CommonAtt, DefEncAtt, empty} binary-linalg.class = vectorproduct | scalarproduct | outerproduct ContExp |= binary-linalg.class vectorproduct = elementvectorproduct
{ CommonAtt, DefEncAtt, empty} scalarproduct = elementscalarproduct
{ CommonAtt, DefEncAtt, empty} outerproduct = elementouterproduct
{ CommonAtt, DefEncAtt, empty} constant-set.class = integers | reals | rationals | naturalnumbers | complexes | primes | emptyset ContExp |= constant-set.class integers = elementintegers
{ CommonAtt, DefEncAtt, empty} reals = elementreals
{ CommonAtt, DefEncAtt, empty} rationals = elementrationals
{ CommonAtt, DefEncAtt, empty} naturalnumbers = elementnaturalnumbers
{ CommonAtt, DefEncAtt, empty} complexes = elementcomplexes
{ CommonAtt, DefEncAtt, empty} primes = elementprimes
{ CommonAtt, DefEncAtt, empty} emptyset = elementemptyset
{ CommonAtt, DefEncAtt, empty} constant-arith.class = exponentiale | imaginaryi | notanumber | true | false | pi | eulergamma | infinity ContExp |= constant-arith.class exponentiale = elementexponentiale
{ CommonAtt, DefEncAtt, empty} imaginaryi = elementimaginaryi
{ CommonAtt, DefEncAtt, empty} notanumber = elementnotanumber
{ CommonAtt, DefEncAtt, empty} true = elementtrue
{ CommonAtt, DefEncAtt, empty} false = elementfalse
{ CommonAtt, DefEncAtt, empty} pi = elementpi
{ CommonAtt, DefEncAtt, empty} eulergamma = elementeulergamma
{ CommonAtt, DefEncAtt, empty} infinity = elementinfinity
{ CommonAtt, DefEncAtt, empty}
Normally, a MathML expression does not constitute an entire XML document. MathML is designed to be used as the mathematics fragment of larger markup languages. In particular it is designed to be used as a module in documents marked up with the XHTML family of markup languages. As RelaxNG directly supports modular development, this is usually very easy: an XHTML+MathML schema can be specified as simply as
# A RelaxNG Schema for XHTML+MathML include "xhtml.rnc" math = external "mathml3.rnc" Inline.class |= math Block.class |= math
assuming that we have access to a modular RelaxNG schema for XHTML that uses
Inline.class
and Block.class
to collect the the content models
for inline and block-level elements.
Customizing the MathML3 schema so that we can restrict the content of
annotation-xml
elements is similarly simple, for example:
# A RelaxNG Schema for MathML with OpenMath3 annotations omobj = external "openmath3.rnc" include "mathml3.rnc" {anotation-xml.model = omobj}
The MathML3 schema is organized so that subsetting to one of the sublanguages specified here is easy. To include strict content MathML3 in a schema just include
include "mathml3-common.rnc" include "mathml3-strict-content.rnc"
instead of include mathml3.rnc
.
For details about RelaxNG grammars and modularization see [RELAX-NG] or [RelaxNGBook].
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 XML Recommendation [Namespaces] allows the prefix to be changed at arbitrary points in the document, since namespace prefixes may be declared on any element.
The MathML DTD uses the strategy outlined in [Modularization] of making 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.
Note that the DTD is far more permissive than the Relax NG schema, many constraints on attribute values may not be enforced using DTD syntax. For example, many attributes expect numerical values or specific micro-syntax, but are declared as CDATA (arbitrary string) in the DTD. However, lack of enforcement of a requirement in the DTD does not 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 Section 2.3.2 Handling of Errors for a description of how MathML renderers should respond to MathML errors.)
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 Chapter 7 Characters, Entities and Fonts first, then Chapter 3 Presentation Markup, Chapter 4 Content Markup, then Section 2.1 MathML Syntax and Grammar, and then other parts of the text.)
MathML documents should be validated using the XML DTD for MathML, http://www.w3.org/Math/DTD/mathml3/mathml3.dtd.
Documents using this DTD should contain a DOCTYPE declaration of the form:
<!DOCTYPE math PUBLIC "-//W3C//DTD MathML 3.0//EN" "http://www.w3.org/Math/DTD/mathml3/mathml3.dtd">
The URI may be changed to that of a local copy of the DTD if required.
MathML fragments can be validated using the XML Schema for MathML, located at http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd. The provided schema has been mechanically generated from the Relax NG schema, it omits some constraints that can not be enforced using XSD syntax.
Similarly to the DOCTYPE declaration used in documents, it is possible to link a MathML fragment to the XML Schema, as shown below.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.w3.org/1998/Math/MathML http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd"> ... </mml:math>
The xsi:schemaLocation
attribute belongs to the XML Schema
instance namespace, defined in [XMLSchemas]. The value
of the attribute is a pair of URIs: the first is the MathML namespace
URI and the second is the location of the schema for that
namespace. The second URI may be changed to that of a local copy if
required.
As the XML Schema specification indicates, the use of the
schemaLocation
attribute is only a hint for schema
processors: validation by a MathML-aware processor can be performed
without requiring that the schemaLocation
attribute be
declared in the instance. Moreover, a processor can even ignore or
override the specified schema URI, if it is directed to.
As noted in Section 6.4.1 Mixing MathML and XHTML The schema used for
XHTML+MathML documents extends the content model of mtext
to
allow inline (phrasing) html elements. The extension used by the W3C
validator uses the RelaxNG schema, however DTD and XSD versions of the
grammar may be similarly extended.
See Section 6.4.3 Mixing MathML and HTML for more details of the parsing of MathML in HTML.
This normative appendix registers three media types for MathML, "application/mathml+xml", "application/mathml-presentation+xml" and "application/mathml-content+xml", in conformance with [RFC4288] and W3CRegMedia. The media-types have been approved by IESG for registration with IANA as announced by IETF and are visible on the list of application media-types.
MathML contains two distinct vocabularies. Presentation markup is
for encoding visual presentation, and consists of the elements defined
in Chapter 3 Presentation Markup. Content markup is for encoding
mathematical meaning, and consists of the elements defined in Chapter 4 Content Markup. In addition, both the presentation and content
vocabularies contain the math
, semantics
,
annotation
and annotation-xml
elements. The MathML
media types should be used as follows:
Some MathML applications may import and export only one of these two vocabularies, while others may produce and consume each in a different way, and still others may process both without any distinction between the two. Internally, many MathML processors favor one vocabulary, and support the other vocabulary via conversion if at all. For example, computational software typically favors content markup while typesetting software generally favors presentation markup. By using separate media types for MathML instances consisting solely of presentation or solely of content markup, such processors can conduct negotiation for MathML representations in the preferred vocabulary. For example, consider two web services offering mathematical computation services such as a spreadsheet and a computer algebra system. Internally both prefer content markup, but by default, both generate presentation markup for output. In the absence of media type negotiation, a likely scenario for an exchange between two systems involves two conversions, content to presentation and back again. With negotiation, the conversions are eliminated. Similarly, a client with a MathML instance in one of the vocabularies might seek a web service that preferred that vocabulary.
MathML is commonly used in compound document settings, e.g. within HTML, where content is drawn from a variety of sources, and processed with multiple tools. In these cases, the composition of MathML expressions generally is not known or at least cannot be guaranteed by a user agent. Consequently, the "application/mathml+xml" type should be used, as it may be applied to any valid MathML expression. Since most applications involve data from untrusted sources, "application/mathml+xml" will commonly be appropriate to use as a default type, and all MathML processors are encouraged to accept it as a fallback to the more specific formats.
The media types described here may be applied to instances of all versions of MathML up to and including MathML 3. MathML instances do not contain version numbers, so processors and producers must follow the normative backward compatibility behavior described in this specification.
This registration has been submitted to community review and has been approved by IESG for registration with IANA.
application
mathml+xml
charset
parameter of
application/xml as specified in [RFC3023]As with other XML types and as noted in [RFC3023] section 10, repeated expansion of maliciously constructed XML entities can be used to consume large amounts of memory, which may cause XML processors in constrained environments to fail.
Several MathML elements may cause arbitrary URIs to be referenced. In this case, the security issues of [RFC3986], section 7, should be considered.
In common with HTML, MathML documents may reference external media such as images, style sheets, and scripting languages. Scripting languages are executable content. In this case, the security considerations in the Media Type registrations for those formats shall apply. Similarly, MathML annotation elements may contain content intended for execution or processing. In the case where the processor recognizes and processes the additional content, or where further processing of that content is dispatched to other processors, additional security issues potentially arise. Since the normative semantics of this specification do not require processing of annotation elements, such issues fall outside the domain of this registration document.
MathML may be used to describe mathematical expressions intended for evaluation in computing systems. Because of the nature of mathematics, a seemingly innocuous expression may lead to a computation which does not terminate or is impractically large. This introduces the risk that computational processors in constrained environments may fail.
In addition, because of the extensibility features for MathML and of XML in general, it is possible that "application/mathml+xml" may describe content that has security implications beyond those described here. However, if the processor follows only the normative semantics of this specification, this content will be outside the MathML namespace and shall be ignored.
This specification describes processing semantics that dictate behavior that must be followed when dealing with, among other things, unrecognized elements and attributes, both in the MathML namespace and in other namespaces.
Because MathML is extensible, conformant "application/mathml+xml" processors must expect that content received is well-formed XML, but it cannot be guaranteed that the content is valid to a particular DTD or Schema or that the processor will recognize all of the elements and attributes in the document.
MathML instances do not contain version numbers, so processors and producers must follow the normative backward compatibility behavior described in this specification.
In computational contexts, the result of evaluating a MathML expression is system-specific, and is not guaranteed to be interoperable between systems.
.mml
MathML
public.mathml
conforming to public.xml
This registration has been submitted to community review and has been approved by IESG for registration with IANA.
application
mathml-presentation+xml
charset
parameter of
application/xml as specified in [RFC3023]As with other XML types and as noted in [RFC3023] section 10, repeated expansion of maliciously constructed XML entities can be used to consume large amounts of memory, which may cause XML processors in constrained environments to fail.
Several MathML elements may cause arbitrary URIs to be referenced. In this case, the security issues of [RFC3986], section 7, should be considered.
In common with HTML, MathML documents may reference external media such as images, style sheets, and scripting languages. Scripting languages are executable content. In this case, the security considerations in the Media Type registrations for those formats shall apply. Similarly, MathML annotation elements may contain content intended for execution or processing. In the case where the processor recognizes and processes the additional content, or where further processing of that content is dispatched to other processors, additional security issues potentially arise. Since the normative semantics of this specification do not require processing of annotation elements, such issues fall outside the domain of this registration document.
MathML may be used to describe mathematical expressions intended for evaluation in computing systems. Because of the nature of mathematics, a seemingly innocuous expression may lead to a computation which does not terminate or is impractically large. This introduces the risk that computational processors in constrained environments may fail.
In addition, because of the extensibility features for MathML and of XML in general, it is possible that "application/mathml-presentation+xml" may describe content that has security implications beyond those described here. However, if the processor follows only the normative semantics of this specification, this content will be outside the MathML namespace and shall be ignored.
This specification describes processing semantics that dictate behavior that must be followed when dealing with, among other things, unrecognized elements and attributes, both in the MathML namespace and in other namespaces.
Because MathML is extensible, conformant "application/mathml-presentation+xml" processors must expect that content received is well-formed XML, but it cannot be guaranteed that the content is valid to a particular DTD or Schema or that the processor will recognize all of the elements and attributes in the document.
MathML instances do not contain version numbers, so processors and producers must follow the normative backward compatibility behavior described in this specification.
In computational contexts, the result of evaluating a MathML expression is system-specific, and is not guaranteed to be interoperable between systems.
This specification does not record a file extension for the media type "application/mathml-presentation+xml" because we expect tools processing files with MathML inside to have sufficient information with the generic media-type (application/mathml+xml) while other content negotiation forms will take advantage of specific media-types.
MathML Presentation
public.mathml.presentation
conforming to
public.mathml
(described above) conforming to public.xml
This registration has been submitted to community review and has been approved by IESG for registration with IANA.
application
mathml-content+xml
charset
parameter of
application/xml as specified in [RFC3023]As with other XML types and as noted in [RFC3023] section 10, repeated expansion of maliciously constructed XML entities can be used to consume large amounts of memory, which may cause XML processors in constrained environments to fail.
Several MathML elements may cause arbitrary URIs to be referenced. In this case, the security issues of [RFC3986], section 7, should be considered.
In common with HTML, MathML documents may reference external media such as images, style sheets, and scripting languages. Scripting languages are executable content. In this case, the security considerations in the Media Type registrations for those formats shall apply. Similarly, MathML annotation elements may contain content intended for execution or processing. In the case where the processor recognizes and processes the additional content, or where further processing of that content is dispatched to other processors, additional security issues potentially arise. Since the normative semantics of this specification do not require processing of annotation elements, such issues fall outside the domain of this registration document.
MathML may be used to describe mathematical expressions intended for evaluation in computing systems. Because of the nature of mathematics, a seemingly innocuous expression may lead to a computation which does not terminate or is impractically large. This introduces the risk that computational processors in constrained environments may fail.
In addition, because of the extensibility features for MathML and of XML in general, it is possible that "application/mathml-content+xml" may describe content that has security implications beyond those described here. However, if the processor follows only the normative semantics of this specification, this content will be outside the MathML namespace and shall be ignored.
This specification describes processing semantics that dictate behavior that must be followed when dealing with, among other things, unrecognized elements and attributes, both in the MathML namespace and in other namespaces.
Because MathML is extensible, conformant "application/mathml-content+xml" processors must expect that content received is well-formed XML, but it cannot be guaranteed that the content is valid to a particular DTD or Schema or that the processor will recognize all of the elements and attributes in the document.
MathML instances do not contain version numbers, so processors and producers must follow the normative backward compatibility behavior described in this specification.
In computational contexts, the result of evaluating a MathML expression is system-specific, and is not guaranteed to be interoperable between systems.
This specification does not record a file extension for the media type "application/mathml-content+xml" because we expect tools processing files with MathML inside to have sufficient information with the generic media-type (application/mathml+xml) while other content negotiation forms will take advantage of specific media-types.
MathML Content
public.mathml.content
conforming to
public.mathml
(described above) conforming to public.xml
The following table gives the suggested dictionary of rendering
properties for operators, fences, separators, and accents in MathML, all of
which are represented by mo
elements. For brevity,
all such elements will be called simply "operators" in this
Appendix.
Note that the dictionary is indexed not just by the element
content, but by the element content and form
attribute
value, together. Operators with more than one possible form have more
than one entry. The MathML specification describes how the renderer
chooses ("infers") which form to use when no form
attribute is given; see Section 3.2.5.7.2 Default value of the form attribute.
Having made that choice, or with the form
attribute
explicitly specified in the <mo>
element's start
tag, the MathML renderer uses the remaining attributes from the
dictionary entry for the appropriate single form of that operator,
ignoring the entries for the other possible forms.
In the table below, all non-ASCII characters are represented by XML-style hexadecimal numeric character references. The choice of markup (character data, numeric character reference or named entity reference) for a given character in MathML has no effect on its rendering.
Each row of the table is indexed as described above by the both the character
(given by code point and Unicode Name) and the value of the form attribute.
The fourth column gives the priority which as described in Section 3.3.1 Horizontally Group Sub-Expressions
<mrow>
),
is significant for the proper grouping of sub-expressions using <mrow>
.
The rule described
there is especially relevant to the automatic generation of MathML by
conversion from other formats for displayed mathematics, such as TEX,
which do not always specify how sub-expressions nest.
The length valued attributes such as lspace
are given explicitly in the following columns.
Boolean valued attributes such as stretchy
are specified together in the final Properties
column by listing the attribute name if its value should be set to true by the dictionary.
Finally some rarely used non-boolean properties are shown in the Properties column together with a value,
for example linebreakstyle
=after.
Any attribute not listed for some entry has its default value,
which is given in parentheses in the table of attributes in Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
.
( | ( | left parenthesis | prefix | 1 | 0 | 0 | fence, stretchy |
---|
could be expressed as an mo
element by:
<mo form="prefix" fence="true" stretchy="true" lspace="0em" rspace="0em"> ( </mo> |
(note the whitespace added around the content for readability; Such whitespace will be ignored by a MathML system, as described in Section 2.1.7 Collapsing Whitespace in Input.
This entry means that, for MathML renderers which use this
suggested operator dictionary, giving the element
<mo form="prefix"> ( </mo>
alone, or simply
<mo> ( </mo>
in a position for which
form="prefix"
would be inferred (see below), is equivalent to giving the element
with all attributes as shown above.
In some versions of this specification, the rows of the table may be reordered by clicking on a heading in the top row, to cause the table to be ordered by that column.
lspace
and
rspace
attributes
The values for lspace
and rspace
given here range from 0 to 7,
they are given numerically in order to save space in the table,
the values should be taken as referring to the named mathematical spaces, as follows.
Table Entry | Named Space | Default Length |
---|---|---|
0 | 0 em | |
1 | veryverythinmathspace | 1/18 em |
2 | verythinmathspace | 2/18 em |
3 | thinmathspace | 3/18 em |
4 | mediummathspace | 4/18 em |
5 | thickmathspace | 5/18 em |
6 | verythickmathspace | 6/18 em |
7 | veryverythickmathspace | 7/18 em |
For the invisible operators whose content is ⁢
or ⁡
,
it is suggested that MathML renderers choose spacing in a context-sensitive
way (which is an exception to the static values given in the following
table). For <mo>⁡</mo>
, the total
spacing ("lspace"+"rspace") in
expressions such as "sin x" (where the right operand
doesn't start with a fence) should be greater than zero; for
<mo>⁢</mo>
, the total spacing
should be greater than zero when both operands (or the nearest tokens on
either side, if on the baseline) are identifiers displayed in a non-slanted
font (i.e.. under the suggested rules, when both operands are
multi-character identifiers).
Some renderers may wish to use no spacing for most operators
appearing in scripts (i.e. when scriptlevel
is greater
than 0; see Section 3.3.4 Style Change <mstyle>
), as is the case in TEX.
Character | Glyph | Name | form | priority | lspace | rspace | Properties |
---|---|---|---|---|---|---|---|
‘ | ‘ | left single quotation mark | prefix | 10 | 0 | 0 | fence |
’ | ’ | right single quotation mark | postfix | 10 | 0 | 0 | fence |
“ | “ | left double quotation mark | prefix | 10 | 0 | 0 | fence |
” | ” | right double quotation mark | postfix | 10 | 0 | 0 | fence |
( | ( | left parenthesis | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
) | ) | right parenthesis | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
[ | [ | left square bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
] | ] | right square bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
{ | { | left curly bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
| | | | vertical line | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
| | | | vertical line | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
|| | || | multiple character operator: || | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
|| | || | multiple character operator: || | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
||| | ||| | multiple character operator: ||| | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
||| | ||| | multiple character operator: ||| | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
} | } | right curly bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
‖ | ‖ | double vertical line | prefix | 20 | 0 | 0 | fence, stretchy |
‖ | ‖ | double vertical line | postfix | 20 | 0 | 0 | fence, stretchy |
⌈ | ⌈ | left ceiling | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⌉ | ⌉ | right ceiling | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⌊ | ⌊ | left floor | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⌋ | ⌋ | right floor | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
〈 | 〈 | left-pointing angle bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
〉 | 〉 | right-pointing angle bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
❲ | ❲ | light left tortoise shell bracket ornament | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
❳ | ❳ | light right tortoise shell bracket ornament | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟦ | ⟦ | mathematical left white square bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟧ | ⟧ | mathematical right white square bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟨ | ⟨ | mathematical left angle bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟩ | ⟩ | mathematical right angle bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟪ | ⟪ | mathematical left double angle bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟫ | ⟫ | mathematical right double angle bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟬ | ⟬ | mathematical left white tortoise shell bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟭ | ⟭ | mathematical right white tortoise shell bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟮ | ⟮ | mathematical left flattened parenthesis | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⟯ | ⟯ | mathematical right flattened parenthesis | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦀ | ⦀ | triple vertical bar delimiter | prefix | 20 | 0 | 0 | fence, stretchy |
⦀ | ⦀ | triple vertical bar delimiter | postfix | 20 | 0 | 0 | fence, stretchy |
⦃ | ⦃ | left white curly bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦄ | ⦄ | right white curly bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦅ | ⦅ | left white parenthesis | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦆ | ⦆ | right white parenthesis | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦇ | ⦇ | z notation left image bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦈ | ⦈ | z notation right image bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦉ | ⦉ | z notation left binding bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦊ | ⦊ | z notation right binding bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦋ | ⦋ | left square bracket with underbar | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦌ | ⦌ | right square bracket with underbar | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦍ | ⦍ | left square bracket with tick in top corner | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦎ | ⦎ | right square bracket with tick in bottom corner | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦏ | ⦏ | left square bracket with tick in bottom corner | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦐ | ⦐ | right square bracket with tick in top corner | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦑ | ⦑ | left angle bracket with dot | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦒ | ⦒ | right angle bracket with dot | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦓ | ⦓ | left arc less-than bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦔ | ⦔ | right arc greater-than bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦕ | ⦕ | double left arc greater-than bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦖ | ⦖ | double right arc less-than bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦗ | ⦗ | left black tortoise shell bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⦘ | ⦘ | right black tortoise shell bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
⧼ | ⧼ | left-pointing curved angle bracket | prefix | 20 | 0 | 0 | fence, stretchy, symmetric |
⧽ | ⧽ | right-pointing curved angle bracket | postfix | 20 | 0 | 0 | fence, stretchy, symmetric |
; | ; | semicolon | infix | 30 | 0 | 3 | separator, linebreakstyle=after |
, | , | comma | infix | 40 | 0 | 3 | separator, linebreakstyle=after |
⁣ | | invisible separator | infix | 40 | 0 | 0 | separator, linebreakstyle=after |
∴ | ∴ | therefore | infix | 70 | 5 | 5 | |
∵ | ∵ | because | infix | 70 | 5 | 5 | |
-> | -> | multiple character operator: -> | infix | 90 | 5 | 5 | |
.. | .. | multiple character operator: .. | postfix | 100 | 0 | 0 | |
... | ... | multiple character operator: ... | postfix | 100 | 0 | 0 | |
: | : | colon | infix | 100 | 1 | 2 | |
϶ | ϶ | greek reversed lunate epsilon symbol | infix | 110 | 5 | 5 | |
… | … | horizontal ellipsis | infix | 150 | 0 | 0 | |
⋮ | ⋮ | vertical ellipsis | infix | 150 | 5 | 5 | |
⋯ | ⋯ | midline horizontal ellipsis | infix | 150 | 0 | 0 | |
⋱ | ⋱ | down right diagonal ellipsis | infix | 150 | 5 | 5 | |
∋ | ∋ | contains as member | infix | 160 | 5 | 5 | |
⊢ | ⊢ | right tack | infix | 170 | 5 | 5 | |
⊣ | ⊣ | left tack | infix | 170 | 5 | 5 | |
⊤ | ⊤ | down tack | infix | 170 | 5 | 5 | |
⊨ | ⊨ | true | infix | 170 | 5 | 5 | |
⊩ | ⊩ | forces | infix | 170 | 5 | 5 | |
⊬ | ⊬ | does not prove | infix | 170 | 5 | 5 | |
⊭ | ⊭ | not true | infix | 170 | 5 | 5 | |
⊮ | ⊮ | does not force | infix | 170 | 5 | 5 | |
⊯ | ⊯ | negated double vertical bar double right turnstile | infix | 170 | 5 | 5 | |
∨ | ∨ | logical or | infix | 190 | 4 | 4 | |
&& | && | multiple character operator: && | infix | 200 | 4 | 4 | |
∧ | ∧ | logical and | infix | 200 | 4 | 4 | |
∀ | ∀ | for all | prefix | 230 | 2 | 1 | |
∃ | ∃ | there exists | prefix | 230 | 2 | 1 | |
∄ | ∄ | there does not exist | prefix | 230 | 2 | 1 | |
∁ | ∁ | complement | infix | 240 | 1 | 2 | |
∈ | ∈ | element of | infix | 240 | 5 | 5 | |
∉ | ∉ | not an element of | infix | 240 | 5 | 5 | |
∌ | ∌ | does not contain as member | infix | 240 | 5 | 5 | |
⊂ | ⊂ | subset of | infix | 240 | 5 | 5 | |
⊂⃒ | ⊂⃒ | subset of with vertical line | infix | 240 | 5 | 5 | |
⊃ | ⊃ | superset of | infix | 240 | 5 | 5 | |
⊃⃒ | ⊃⃒ | superset of with vertical line | infix | 240 | 5 | 5 | |
⊄ | ⊄ | not a subset of | infix | 240 | 5 | 5 | |
⊅ | ⊅ | not a superset of | infix | 240 | 5 | 5 | |
⊆ | ⊆ | subset of or equal to | infix | 240 | 5 | 5 | |
⊇ | ⊇ | superset of or equal to | infix | 240 | 5 | 5 | |
⊈ | ⊈ | neither a subset of nor equal to | infix | 240 | 5 | 5 | |
⊉ | ⊉ | neither a superset of nor equal to | infix | 240 | 5 | 5 | |
⊊ | ⊊ | subset of with not equal to | infix | 240 | 5 | 5 | |
⊋ | ⊋ | superset of with not equal to | infix | 240 | 5 | 5 | |
<= | <= | multiple character operator: <= | infix | 241 | 5 | 5 | |
≤ | ≤ | less-than or equal to | infix | 241 | 5 | 5 | |
≥ | ≥ | greater-than or equal to | infix | 242 | 5 | 5 | |
> | > | greater-than sign | infix | 243 | 5 | 5 | |
>= | >= | multiple character operator: >= | infix | 243 | 5 | 5 | |
≯ | ≯ | not greater-than | infix | 244 | 5 | 5 | |
< | < | less-than sign | infix | 245 | 5 | 5 | |
≮ | ≮ | not less-than | infix | 246 | 5 | 5 | |
≈ | ≈ | almost equal to | infix | 247 | 5 | 5 | |
∼ | ∼ | tilde operator | infix | 250 | 5 | 5 | |
≉ | ≉ | not almost equal to | infix | 250 | 5 | 5 | |
≢ | ≢ | not identical to | infix | 252 | 5 | 5 | |
≠ | ≠ | not equal to | infix | 255 | 5 | 5 | |
!= | != | multiple character operator: != | infix | 260 | 4 | 4 | |
*= | *= | multiple character operator: *= | infix | 260 | 4 | 4 | |
+= | += | multiple character operator: += | infix | 260 | 4 | 4 | |
-= | -= | multiple character operator: -= | infix | 260 | 4 | 4 | |
/= | /= | multiple character operator: /= | infix | 260 | 4 | 4 | |
:= | := | multiple character operator: := | infix | 260 | 4 | 4 | |
= | = | equals sign | infix | 260 | 5 | 5 | |
== | == | multiple character operator: == | infix | 260 | 4 | 4 | |
∝ | ∝ | proportional to | infix | 260 | 5 | 5 | |
∤ | ∤ | does not divide | infix | 260 | 5 | 5 | |
∥ | ∥ | parallel to | infix | 260 | 5 | 5 | |
∦ | ∦ | not parallel to | infix | 260 | 5 | 5 | |
≁ | ≁ | not tilde | infix | 260 | 5 | 5 | |
≃ | ≃ | asymptotically equal to | infix | 260 | 5 | 5 | |
≄ | ≄ | not asymptotically equal to | infix | 260 | 5 | 5 | |
≅ | ≅ | approximately equal to | infix | 260 | 5 | 5 | |
≆ | ≆ | approximately but not actually equal to | infix | 260 | 5 | 5 | |
≇ | ≇ | neither approximately nor actually equal to | infix | 260 | 5 | 5 | |
≍ | ≍ | equivalent to | infix | 260 | 5 | 5 | |
≔ | ≔ | colon equals | infix | 260 | 5 | 5 | |
≗ | ≗ | ring equal to | infix | 260 | 5 | 5 | |
≙ | ≙ | estimates | infix | 260 | 5 | 5 | |
≚ | ≚ | equiangular to | infix | 260 | 5 | 5 | |
≛ | ≛ | star equals | infix | 260 | 5 | 5 | |
≜ | ≜ | delta equal to | infix | 260 | 5 | 5 | |
≟ | ≟ | questioned equal to | infix | 260 | 5 | 5 | |
≡ | ≡ | identical to | infix | 260 | 5 | 5 | |
≨ | ≨ | less-than but not equal to | infix | 260 | 5 | 5 | |
≩ | ≩ | greater-than but not equal to | infix | 260 | 5 | 5 | |
≪ | ≪ | much less-than | infix | 260 | 5 | 5 | |
≪̸ | ≪̸ | much less than with slash | infix | 260 | 5 | 5 | |
≫ | ≫ | much greater-than | infix | 260 | 5 | 5 | |
≫̸ | ≫̸ | much greater than with slash | infix | 260 | 5 | 5 | |
≭ | ≭ | not equivalent to | infix | 260 | 5 | 5 | |
≰ | ≰ | neither less-than nor equal to | infix | 260 | 5 | 5 | |
≱ | ≱ | neither greater-than nor equal to | infix | 260 | 5 | 5 | |
≺ | ≺ | precedes | infix | 260 | 5 | 5 | |
≻ | ≻ | succeeds | infix | 260 | 5 | 5 | |
≼ | ≼ | precedes or equal to | infix | 260 | 5 | 5 | |
≽ | ≽ | succeeds or equal to | infix | 260 | 5 | 5 | |
⊀ | ⊀ | does not precede | infix | 260 | 5 | 5 | |
⊁ | ⊁ | does not succeed | infix | 260 | 5 | 5 | |
⊥ | ⊥ | up tack | infix | 260 | 5 | 5 | |
⊴ | ⊴ | normal subgroup of or equal to | infix | 260 | 5 | 5 | |
⊵ | ⊵ | contains as normal subgroup or equal to | infix | 260 | 5 | 5 | |
⋉ | ⋉ | left normal factor semidirect product | infix | 260 | 4 | 4 | |
⋊ | ⋊ | right normal factor semidirect product | infix | 260 | 4 | 4 | |
⋋ | ⋋ | left semidirect product | infix | 260 | 4 | 4 | |
⋌ | ⋌ | right semidirect product | infix | 260 | 4 | 4 | |
⋔ | ⋔ | pitchfork | infix | 260 | 5 | 5 | |
⋖ | ⋖ | less-than with dot | infix | 260 | 5 | 5 | |
⋗ | ⋗ | greater-than with dot | infix | 260 | 5 | 5 | |
⋘ | ⋘ | very much less-than | infix | 260 | 5 | 5 | |
⋙ | ⋙ | very much greater-than | infix | 260 | 5 | 5 | |
⋪ | ⋪ | not normal subgroup of | infix | 260 | 5 | 5 | |
⋫ | ⋫ | does not contain as normal subgroup | infix | 260 | 5 | 5 | |
⋬ | ⋬ | not normal subgroup of or equal to | infix | 260 | 5 | 5 | |
⋭ | ⋭ | does not contain as normal subgroup or equal | infix | 260 | 5 | 5 | |
■ | ■ | black square | infix | 260 | 3 | 3 | |
□ | □ | white square | infix | 260 | 3 | 3 | |
▪ | ▪ | black small square | infix | 260 | 3 | 3 | |
▫ | ▫ | white small square | infix | 260 | 3 | 3 | |
▭ | ▭ | white rectangle | infix | 260 | 3 | 3 | |
▮ | ▮ | black vertical rectangle | infix | 260 | 3 | 3 | |
▯ | ▯ | white vertical rectangle | infix | 260 | 3 | 3 | |
▰ | ▰ | black parallelogram | infix | 260 | 3 | 3 | |
▱ | ▱ | white parallelogram | infix | 260 | 3 | 3 | |
△ | △ | white up-pointing triangle | infix | 260 | 4 | 4 | |
▴ | ▴ | black up-pointing small triangle | infix | 260 | 4 | 4 | |
▵ | ▵ | white up-pointing small triangle | infix | 260 | 4 | 4 | |
▶ | ▶ | black right-pointing triangle | infix | 260 | 4 | 4 | |
▷ | ▷ | white right-pointing triangle | infix | 260 | 4 | 4 | |
▸ | ▸ | black right-pointing small triangle | infix | 260 | 4 | 4 | |
▹ | ▹ | white right-pointing small triangle | infix | 260 | 4 | 4 | |
▼ | ▼ | black down-pointing triangle | infix | 260 | 4 | 4 | |
▽ | ▽ | white down-pointing triangle | infix | 260 | 4 | 4 | |
▾ | ▾ | black down-pointing small triangle | infix | 260 | 4 | 4 | |
▿ | ▿ | white down-pointing small triangle | infix | 260 | 4 | 4 | |
◀ | ◀ | black left-pointing triangle | infix | 260 | 4 | 4 | |
◁ | ◁ | white left-pointing triangle | infix | 260 | 4 | 4 | |
◂ | ◂ | black left-pointing small triangle | infix | 260 | 4 | 4 | |
◃ | ◃ | white left-pointing small triangle | infix | 260 | 4 | 4 | |
◄ | ◄ | black left-pointing pointer | infix | 260 | 4 | 4 | |
◅ | ◅ | white left-pointing pointer | infix | 260 | 4 | 4 | |
◆ | ◆ | black diamond | infix | 260 | 4 | 4 | |
◇ | ◇ | white diamond | infix | 260 | 4 | 4 | |
◈ | ◈ | white diamond containing black small diamond | infix | 260 | 4 | 4 | |
◉ | ◉ | fisheye | infix | 260 | 4 | 4 | |
◌ | ◌ | dotted circle | infix | 260 | 4 | 4 | |
◍ | ◍ | circle with vertical fill | infix | 260 | 4 | 4 | |
◎ | ◎ | bullseye | infix | 260 | 4 | 4 | |
● | ● | black circle | infix | 260 | 4 | 4 | |
◖ | ◖ | left half black circle | infix | 260 | 4 | 4 | |
◗ | ◗ | right half black circle | infix | 260 | 4 | 4 | |
◦ | ◦ | white bullet | infix | 260 | 4 | 4 | |
⧀ | ⧀ | circled less-than | infix | 260 | 5 | 5 | |
⧁ | ⧁ | circled greater-than | infix | 260 | 5 | 5 | |
⧣ | ⧣ | equals sign and slanted parallel | infix | 260 | 5 | 5 | |
⧤ | ⧤ | equals sign and slanted parallel with tilde above | infix | 260 | 5 | 5 | |
⧥ | ⧥ | identical to and slanted parallel | infix | 260 | 5 | 5 | |
⧦ | ⧦ | gleich stark | infix | 260 | 5 | 5 | |
⧳ | ⧳ | error-barred black circle | infix | 260 | 3 | 3 | |
⪇ | ⪇ | less-than and single-line not equal to | infix | 260 | 5 | 5 | |
⪈ | ⪈ | greater-than and single-line not equal to | infix | 260 | 5 | 5 | |
⪯ | ⪯ | precedes above single-line equals sign | infix | 260 | 5 | 5 | |
⪯̸ | ⪯̸ | precedes above single-line equals sign with slash | infix | 260 | 5 | 5 | |
⪰ | ⪰ | succeeds above single-line equals sign | infix | 260 | 5 | 5 | |
⪰̸ | ⪰̸ | succeeds above single-line equals sign with slash | infix | 260 | 5 | 5 | |
⁄ | ⁄ | fraction slash | infix | 265 | 4 | 4 | stretchy |
∆ | ∆ | increment | infix | 265 | 3 | 3 | |
∊ | ∊ | small element of | infix | 265 | 5 | 5 | |
∍ | ∍ | small contains as member | infix | 265 | 5 | 5 | |
∎ | ∎ | end of proof | infix | 265 | 3 | 3 | |
∕ | ∕ | division slash | infix | 265 | 4 | 4 | stretchy |
∗ | ∗ | asterisk operator | infix | 265 | 4 | 4 | |
∘ | ∘ | ring operator | infix | 265 | 4 | 4 | |
∙ | ∙ | bullet operator | infix | 265 | 4 | 4 | |
∟ | ∟ | right angle | infix | 265 | 5 | 5 | |
∣ | ∣ | divides | infix | 265 | 5 | 5 | |
∶ | ∶ | ratio | infix | 265 | 5 | 5 | |
∷ | ∷ | proportion | infix | 265 | 5 | 5 | |
∸ | ∸ | dot minus | infix | 265 | 4 | 4 | |
∹ | ∹ | excess | infix | 265 | 5 | 5 | |
∺ | ∺ | geometric proportion | infix | 265 | 4 | 4 | |
∻ | ∻ | homothetic | infix | 265 | 5 | 5 | |
∽ | ∽ | reversed tilde | infix | 265 | 5 | 5 | |
∽̱ | ∽̱ | reversed tilde with underline | infix | 265 | 3 | 3 | |
∾ | ∾ | inverted lazy s | infix | 265 | 5 | 5 | |
∿ | ∿ | sine wave | infix | 265 | 3 | 3 | |
≂ | ≂ | minus tilde | infix | 265 | 5 | 5 | |
≂̸ | ≂̸ | minus tilde with slash | infix | 265 | 5 | 5 | |
≊ | ≊ | almost equal or equal to | infix | 265 | 5 | 5 | |
≋ | ≋ | triple tilde | infix | 265 | 5 | 5 | |
≌ | ≌ | all equal to | infix | 265 | 5 | 5 | |
≎ | ≎ | geometrically equivalent to | infix | 265 | 5 | 5 | |
≎̸ | ≎̸ | geometrically equivalent to with slash | infix | 265 | 5 | 5 | |
≏ | ≏ | difference between | infix | 265 | 5 | 5 | |
≏̸ | ≏̸ | difference between with slash | infix | 265 | 5 | 5 | |
≐ | ≐ | approaches the limit | infix | 265 | 5 | 5 | |
≑ | ≑ | geometrically equal to | infix | 265 | 5 | 5 | |
≒ | ≒ | approximately equal to or the image of | infix | 265 | 5 | 5 | |
≓ | ≓ | image of or approximately equal to | infix | 265 | 5 | 5 | |
≕ | ≕ | equals colon | infix | 265 | 5 | 5 | |
≖ | ≖ | ring in equal to | infix | 265 | 5 | 5 | |
≘ | ≘ | corresponds to | infix | 265 | 5 | 5 | |
≝ | ≝ | equal to by definition | infix | 265 | 5 | 5 | |
≞ | ≞ | measured by | infix | 265 | 5 | 5 | |
≣ | ≣ | strictly equivalent to | infix | 265 | 5 | 5 | |
≦ | ≦ | less-than over equal to | infix | 265 | 5 | 5 | |
≦̸ | ≦̸ | less-than over equal to with slash | infix | 265 | 5 | 5 | |
≧ | ≧ | greater-than over equal to | infix | 265 | 5 | 5 | |
≬ | ≬ | between | infix | 265 | 5 | 5 | |
≲ | ≲ | less-than or equivalent to | infix | 265 | 5 | 5 | |
≳ | ≳ | greater-than or equivalent to | infix | 265 | 5 | 5 | |
≴ | ≴ | neither less-than nor equivalent to | infix | 265 | 5 | 5 | |
≵ | ≵ | neither greater-than nor equivalent to | infix | 265 | 5 | 5 | |
≶ | ≶ | less-than or greater-than | infix | 265 | 5 | 5 | |
≷ | ≷ | greater-than or less-than | infix | 265 | 5 | 5 | |
≸ | ≸ | neither less-than nor greater-than | infix | 265 | 5 | 5 | |
≹ | ≹ | neither greater-than nor less-than | infix | 265 | 5 | 5 | |
≾ | ≾ | precedes or equivalent to | infix | 265 | 5 | 5 | |
≿ | ≿ | succeeds or equivalent to | infix | 265 | 5 | 5 | |
≿̸ | ≿̸ | succeeds or equivalent to with slash | infix | 265 | 5 | 5 | |
⊌ | ⊌ | multiset | infix | 265 | 4 | 4 | |
⊍ | ⊍ | multiset multiplication | infix | 265 | 4 | 4 | |
⊎ | ⊎ | multiset union | infix | 265 | 4 | 4 | |
⊏ | ⊏ | square image of | infix | 265 | 5 | 5 | |
⊏̸ | ⊏̸ | square image of with slash | infix | 265 | 5 | 5 | |
⊐ | ⊐ | square original of | infix | 265 | 5 | 5 | |
⊐̸ | ⊐̸ | square original of with slash | infix | 265 | 5 | 5 | |
⊑ | ⊑ | square image of or equal to | infix | 265 | 5 | 5 | |
⊒ | ⊒ | square original of or equal to | infix | 265 | 5 | 5 | |
⊓ | ⊓ | square cap | infix | 265 | 4 | 4 | |
⊔ | ⊔ | square cup | infix | 265 | 4 | 4 | |
⊚ | ⊚ | circled ring operator | infix | 265 | 4 | 4 | |
⊛ | ⊛ | circled asterisk operator | infix | 265 | 4 | 4 | |
⊜ | ⊜ | circled equals | infix | 265 | 4 | 4 | |
⊝ | ⊝ | circled dash | infix | 265 | 4 | 4 | |
⊦ | ⊦ | assertion | infix | 265 | 5 | 5 | |
⊧ | ⊧ | models | infix | 265 | 5 | 5 | |
⊪ | ⊪ | triple vertical bar right turnstile | infix | 265 | 5 | 5 | |
⊫ | ⊫ | double vertical bar double right turnstile | infix | 265 | 5 | 5 | |
⊰ | ⊰ | precedes under relation | infix | 265 | 5 | 5 | |
⊱ | ⊱ | succeeds under relation | infix | 265 | 5 | 5 | |
⊲ | ⊲ | normal subgroup of | infix | 265 | 5 | 5 | |
⊳ | ⊳ | contains as normal subgroup | infix | 265 | 5 | 5 | |
⊶ | ⊶ | original of | infix | 265 | 5 | 5 | |
⊷ | ⊷ | image of | infix | 265 | 5 | 5 | |
⊹ | ⊹ | hermitian conjugate matrix | infix | 265 | 5 | 5 | |
⊺ | ⊺ | intercalate | infix | 265 | 4 | 4 | |
⊻ | ⊻ | xor | infix | 265 | 4 | 4 | |
⊼ | ⊼ | nand | infix | 265 | 4 | 4 | |
⊽ | ⊽ | nor | infix | 265 | 4 | 4 | |
⊾ | ⊾ | right angle with arc | infix | 265 | 3 | 3 | |
⊿ | ⊿ | right triangle | infix | 265 | 3 | 3 | |
⋄ | ⋄ | diamond operator | infix | 265 | 4 | 4 | |
⋆ | ⋆ | star operator | infix | 265 | 4 | 4 | |
⋇ | ⋇ | division times | infix | 265 | 4 | 4 | |
⋈ | ⋈ | bowtie | infix | 265 | 5 | 5 | |
⋍ | ⋍ | reversed tilde equals | infix | 265 | 5 | 5 | |
⋎ | ⋎ | curly logical or | infix | 265 | 4 | 4 | |
⋏ | ⋏ | curly logical and | infix | 265 | 4 | 4 | |
⋐ | ⋐ | double subset | infix | 265 | 5 | 5 | |
⋑ | ⋑ | double superset | infix | 265 | 5 | 5 | |
⋒ | ⋒ | double intersection | infix | 265 | 4 | 4 | |
⋓ | ⋓ | double union | infix | 265 | 4 | 4 | |
⋕ | ⋕ | equal and parallel to | infix | 265 | 5 | 5 | |
⋚ | ⋚ | less-than equal to or greater-than | infix | 265 | 5 | 5 | |
⋛ | ⋛ | greater-than equal to or less-than | infix | 265 | 5 | 5 | |
⋜ | ⋜ | equal to or less-than | infix | 265 | 5 | 5 | |
⋝ | ⋝ | equal to or greater-than | infix | 265 | 5 | 5 | |
⋞ | ⋞ | equal to or precedes | infix | 265 | 5 | 5 | |
⋟ | ⋟ | equal to or succeeds | infix | 265 | 5 | 5 | |
⋠ | ⋠ | does not precede or equal | infix | 265 | 5 | 5 | |
⋡ | ⋡ | does not succeed or equal | infix | 265 | 5 | 5 | |
⋢ | ⋢ | not square image of or equal to | infix | 265 | 5 | 5 | |
⋣ | ⋣ | not square original of or equal to | infix | 265 | 5 | 5 | |
⋤ | ⋤ | square image of or not equal to | infix | 265 | 5 | 5 | |
⋥ | ⋥ | square original of or not equal to | infix | 265 | 5 | 5 | |
⋦ | ⋦ | less-than but not equivalent to | infix | 265 | 5 | 5 | |
⋧ | ⋧ | greater-than but not equivalent to | infix | 265 | 5 | 5 | |
⋨ | ⋨ | precedes but not equivalent to | infix | 265 | 5 | 5 | |
⋩ | ⋩ | succeeds but not equivalent to | infix | 265 | 5 | 5 | |
⋰ | ⋰ | up right diagonal ellipsis | infix | 265 | 5 | 5 | |
⋲ | ⋲ | element of with long horizontal stroke | infix | 265 | 5 | 5 | |
⋳ | ⋳ | element of with vertical bar at end of horizontal stroke | infix | 265 | 5 | 5 | |
⋴ | ⋴ | small element of with vertical bar at end of horizontal stroke | infix | 265 | 5 | 5 | |
⋵ | ⋵ | element of with dot above | infix | 265 | 5 | 5 | |
⋶ | ⋶ | element of with overbar | infix | 265 | 5 | 5 | |
⋷ | ⋷ | small element of with overbar | infix | 265 | 5 | 5 | |
⋸ | ⋸ | element of with underbar | infix | 265 | 5 | 5 | |
⋹ | ⋹ | element of with two horizontal strokes | infix | 265 | 5 | 5 | |
⋺ | ⋺ | contains with long horizontal stroke | infix | 265 | 5 | 5 | |
⋻ | ⋻ | contains with vertical bar at end of horizontal stroke | infix | 265 | 5 | 5 | |
⋼ | ⋼ | small contains with vertical bar at end of horizontal stroke | infix | 265 | 5 | 5 | |
⋽ | ⋽ | contains with overbar | infix | 265 | 5 | 5 | |
⋾ | ⋾ | small contains with overbar | infix | 265 | 5 | 5 | |
⋿ | ⋿ | z notation bag membership | infix | 265 | 5 | 5 | |
▲ | ▲ | black up-pointing triangle | infix | 265 | 4 | 4 | |
❘ | ❘ | light vertical bar | infix | 265 | 5 | 5 | |
⦁ | ⦁ | z notation spot | infix | 265 | 3 | 3 | |
⦂ | ⦂ | z notation type colon | infix | 265 | 3 | 3 | |
⦠ | ⦠ | spherical angle opening left | infix | 265 | 3 | 3 | |
⦡ | ⦡ | spherical angle opening up | infix | 265 | 3 | 3 | |
⦢ | ⦢ | turned angle | infix | 265 | 3 | 3 | |
⦣ | ⦣ | reversed angle | infix | 265 | 3 | 3 | |
⦤ | ⦤ | angle with underbar | infix | 265 | 3 | 3 | |
⦥ | ⦥ | reversed angle with underbar | infix | 265 | 3 | 3 | |
⦦ | ⦦ | oblique angle opening up | infix | 265 | 3 | 3 | |
⦧ | ⦧ | oblique angle opening down | infix | 265 | 3 | 3 | |
⦨ | ⦨ | measured angle with open arm ending in arrow pointing up and right | infix | 265 | 3 | 3 | |
⦩ | ⦩ | measured angle with open arm ending in arrow pointing up and left | infix | 265 | 3 | 3 | |
⦪ | ⦪ | measured angle with open arm ending in arrow pointing down and right | infix | 265 | 3 | 3 | |
⦫ | ⦫ | measured angle with open arm ending in arrow pointing down and left | infix | 265 | 3 | 3 | |
⦬ | ⦬ | measured angle with open arm ending in arrow pointing right and up | infix | 265 | 3 | 3 | |
⦭ | ⦭ | measured angle with open arm ending in arrow pointing left and up | infix | 265 | 3 | 3 | |
⦮ | ⦮ | measured angle with open arm ending in arrow pointing right and down | infix | 265 | 3 | 3 | |
⦯ | ⦯ | measured angle with open arm ending in arrow pointing left and down | infix | 265 | 3 | 3 | |
⦰ | ⦰ | reversed empty set | infix | 265 | 3 | 3 | |
⦱ | ⦱ | empty set with overbar | infix | 265 | 3 | 3 | |
⦲ | ⦲ | empty set with small circle above | infix | 265 | 3 | 3 | |
⦳ | ⦳ | empty set with right arrow above | infix | 265 | 3 | 3 | |
⦴ | ⦴ | empty set with left arrow above | infix | 265 | 3 | 3 | |
⦵ | ⦵ | circle with horizontal bar | infix | 265 | 3 | 3 | |
⦶ | ⦶ | circled vertical bar | infix | 265 | 4 | 4 | |
⦷ | ⦷ | circled parallel | infix | 265 | 4 | 4 | |
⦸ | ⦸ | circled reverse solidus | infix | 265 | 4 | 4 | |
⦹ | ⦹ | circled perpendicular | infix | 265 | 4 | 4 | |
⦺ | ⦺ | circle divided by horizontal bar and top half divided by vertical bar | infix | 265 | 4 | 4 | |
⦻ | ⦻ | circle with superimposed x | infix | 265 | 4 | 4 | |
⦼ | ⦼ | circled anticlockwise-rotated division sign | infix | 265 | 4 | 4 | |
⦽ | ⦽ | up arrow through circle | infix | 265 | 4 | 4 | |
⦾ | ⦾ | circled white bullet | infix | 265 | 4 | 4 | |
⦿ | ⦿ | circled bullet | infix | 265 | 4 | 4 | |
⧂ | ⧂ | circle with small circle to the right | infix | 265 | 3 | 3 | |
⧃ | ⧃ | circle with two horizontal strokes to the right | infix | 265 | 3 | 3 | |
⧄ | ⧄ | squared rising diagonal slash | infix | 265 | 4 | 4 | |
⧅ | ⧅ | squared falling diagonal slash | infix | 265 | 4 | 4 | |
⧆ | ⧆ | squared asterisk | infix | 265 | 4 | 4 | |
⧇ | ⧇ | squared small circle | infix | 265 | 4 | 4 | |
⧈ | ⧈ | squared square | infix | 265 | 4 | 4 | |
⧉ | ⧉ | two joined squares | infix | 265 | 3 | 3 | |
⧊ | ⧊ | triangle with dot above | infix | 265 | 3 | 3 | |
⧋ | ⧋ | triangle with underbar | infix | 265 | 3 | 3 | |
⧌ | ⧌ | s in triangle | infix | 265 | 3 | 3 | |
⧍ | ⧍ | triangle with serifs at bottom | infix | 265 | 3 | 3 | |
⧎ | ⧎ | right triangle above left triangle | infix | 265 | 5 | 5 | |
⧏ | ⧏ | left triangle beside vertical bar | infix | 265 | 5 | 5 | |
⧏̸ | ⧏̸ | left triangle beside vertical bar with slash | infix | 265 | 5 | 5 | |
⧐ | ⧐ | vertical bar beside right triangle | infix | 265 | 5 | 5 | |
⧐̸ | ⧐̸ | vertical bar beside right triangle with slash | infix | 265 | 5 | 5 | |
⧑ | ⧑ | bowtie with left half black | infix | 265 | 5 | 5 | |
⧒ | ⧒ | bowtie with right half black | infix | 265 | 5 | 5 | |
⧓ | ⧓ | black bowtie | infix | 265 | 5 | 5 | |
⧔ | ⧔ | times with left half black | infix | 265 | 5 | 5 | |
⧕ | ⧕ | times with right half black | infix | 265 | 5 | 5 | |
⧖ | ⧖ | white hourglass | infix | 265 | 4 | 4 | |
⧗ | ⧗ | black hourglass | infix | 265 | 4 | 4 | |
⧘ | ⧘ | left wiggly fence | infix | 265 | 3 | 3 | |
⧙ | ⧙ | right wiggly fence | infix | 265 | 3 | 3 | |
⧛ | ⧛ | right double wiggly fence | infix | 265 | 3 | 3 | |
⧜ | ⧜ | incomplete infinity | infix | 265 | 3 | 3 | |
⧝ | ⧝ | tie over infinity | infix | 265 | 3 | 3 | |
⧞ | ⧞ | infinity negated with vertical bar | infix | 265 | 5 | 5 | |
⧠ | ⧠ | square with contoured outline | infix | 265 | 3 | 3 | |
⧡ | ⧡ | increases as | infix | 265 | 5 | 5 | |
⧢ | ⧢ | shuffle product | infix | 265 | 4 | 4 | |
⧧ | ⧧ | thermodynamic | infix | 265 | 3 | 3 | |
⧨ | ⧨ | down-pointing triangle with left half black | infix | 265 | 3 | 3 | |
⧩ | ⧩ | down-pointing triangle with right half black | infix | 265 | 3 | 3 | |
⧪ | ⧪ | black diamond with down arrow | infix | 265 | 3 | 3 | |
⧫ | ⧫ | black lozenge | infix | 265 | 3 | 3 | |
⧬ | ⧬ | white circle with down arrow | infix | 265 | 3 | 3 | |
⧭ | ⧭ | black circle with down arrow | infix | 265 | 3 | 3 | |
⧮ | ⧮ | error-barred white square | infix | 265 | 3 | 3 | |
⧰ | ⧰ | error-barred white diamond | infix | 265 | 3 | 3 | |
⧱ | ⧱ | error-barred black diamond | infix | 265 | 3 | 3 | |
⧲ | ⧲ | error-barred white circle | infix | 265 | 3 | 3 | |
⧵ | ⧵ | reverse solidus operator | infix | 265 | 4 | 4 | |
⧶ | ⧶ | solidus with overbar | infix | 265 | 4 | 4 | |
⧷ | ⧷ | reverse solidus with horizontal stroke | infix | 265 | 4 | 4 | |
⧸ | ⧸ | big solidus | infix | 265 | 3 | 3 | |
⧹ | ⧹ | big reverse solidus | infix | 265 | 3 | 3 | |
⧺ | ⧺ | double plus | infix | 265 | 3 | 3 | |
⧻ | ⧻ | triple plus | infix | 265 | 3 | 3 | |
⧾ | ⧾ | tiny | infix | 265 | 4 | 4 | |
⧿ | ⧿ | miny | infix | 265 | 4 | 4 | |
⨝ | ⨝ | join | infix | 265 | 3 | 3 | |
⨞ | ⨞ | large left triangle operator | infix | 265 | 3 | 3 | |
⨟ | ⨟ | z notation schema composition | infix | 265 | 3 | 3 | |
⨠ | ⨠ | z notation schema piping | infix | 265 | 3 | 3 | |
⨡ | ⨡ | z notation schema projection | infix | 265 | 3 | 3 | |
⨢ | ⨢ | plus sign with small circle above | infix | 265 | 4 | 4 | |
⨣ | ⨣ | plus sign with circumflex accent above | infix | 265 | 4 | 4 | |
⨤ | ⨤ | plus sign with tilde above | infix | 265 | 4 | 4 | |
⨥ | ⨥ | plus sign with dot below | infix | 265 | 4 | 4 | |
⨦ | ⨦ | plus sign with tilde below | infix | 265 | 4 | 4 | |
⨧ | ⨧ | plus sign with subscript two | infix | 265 | 4 | 4 | |
⨨ | ⨨ | plus sign with black triangle | infix | 265 | 4 | 4 | |
⨩ | ⨩ | minus sign with comma above | infix | 265 | 4 | 4 | |
⨪ | ⨪ | minus sign with dot below | infix | 265 | 4 | 4 | |
⨫ | ⨫ | minus sign with falling dots | infix | 265 | 4 | 4 | |
⨬ | ⨬ | minus sign with rising dots | infix | 265 | 4 | 4 | |
⨭ | ⨭ | plus sign in left half circle | infix | 265 | 4 | 4 | |
⨮ | ⨮ | plus sign in right half circle | infix | 265 | 4 | 4 | |
⨰ | ⨰ | multiplication sign with dot above | infix | 265 | 4 | 4 | |
⨱ | ⨱ | multiplication sign with underbar | infix | 265 | 4 | 4 | |
⨲ | ⨲ | semidirect product with bottom closed | infix | 265 | 4 | 4 | |
⨳ | ⨳ | smash product | infix | 265 | 4 | 4 | |
⨴ | ⨴ | multiplication sign in left half circle | infix | 265 | 4 | 4 | |
⨵ | ⨵ | multiplication sign in right half circle | infix | 265 | 4 | 4 | |
⨶ | ⨶ | circled multiplication sign with circumflex accent | infix | 265 | 4 | 4 | |
⨷ | ⨷ | multiplication sign in double circle | infix | 265 | 4 | 4 | |
⨸ | ⨸ | circled division sign | infix | 265 | 4 | 4 | |
⨹ | ⨹ | plus sign in triangle | infix | 265 | 4 | 4 | |
⨺ | ⨺ | minus sign in triangle | infix | 265 | 4 | 4 | |
⨻ | ⨻ | multiplication sign in triangle | infix | 265 | 4 | 4 | |
⨼ | ⨼ | interior product | infix | 265 | 4 | 4 | |
⨽ | ⨽ | righthand interior product | infix | 265 | 4 | 4 | |
⨾ | ⨾ | z notation relational composition | infix | 265 | 4 | 4 | |
⩀ | ⩀ | intersection with dot | infix | 265 | 4 | 4 | |
⩁ | ⩁ | union with minus sign | infix | 265 | 4 | 4 | |
⩂ | ⩂ | union with overbar | infix | 265 | 4 | 4 | |
⩃ | ⩃ | intersection with overbar | infix | 265 | 4 | 4 | |
⩄ | ⩄ | intersection with logical and | infix | 265 | 4 | 4 | |
⩅ | ⩅ | union with logical or | infix | 265 | 4 | 4 | |
⩆ | ⩆ | union above intersection | infix | 265 | 4 | 4 | |
⩇ | ⩇ | intersection above union | infix | 265 | 4 | 4 | |
⩈ | ⩈ | union above bar above intersection | infix | 265 | 4 | 4 | |
⩉ | ⩉ | intersection above bar above union | infix | 265 | 4 | 4 | |
⩊ | ⩊ | union beside and joined with union | infix | 265 | 4 | 4 | |
⩋ | ⩋ | intersection beside and joined with intersection | infix | 265 | 4 | 4 | |
⩌ | ⩌ | closed union with serifs | infix | 265 | 4 | 4 | |
⩍ | ⩍ | closed intersection with serifs | infix | 265 | 4 | 4 | |
⩎ | ⩎ | double square intersection | infix | 265 | 4 | 4 | |
⩏ | ⩏ | double square union | infix | 265 | 4 | 4 | |
⩐ | ⩐ | closed union with serifs and smash product | infix | 265 | 4 | 4 | |
⩑ | ⩑ | logical and with dot above | infix | 265 | 4 | 4 | |
⩒ | ⩒ | logical or with dot above | infix | 265 | 4 | 4 | |
⩓ | ⩓ | double logical and | infix | 265 | 4 | 4 | |
⩔ | ⩔ | double logical or | infix | 265 | 4 | 4 | |
⩕ | ⩕ | two intersecting logical and | infix | 265 | 4 | 4 | |
⩖ | ⩖ | two intersecting logical or | infix | 265 | 4 | 4 | |
⩗ | ⩗ | sloping large or | infix | 265 | 4 | 4 | |
⩘ | ⩘ | sloping large and | infix | 265 | 4 | 4 | |
⩙ | ⩙ | logical or overlapping logical and | infix | 265 | 5 | 5 | |
⩚ | ⩚ | logical and with middle stem | infix | 265 | 4 | 4 | |
⩛ | ⩛ | logical or with middle stem | infix | 265 | 4 | 4 | |
⩜ | ⩜ | logical and with horizontal dash | infix | 265 | 4 | 4 | |
⩝ | ⩝ | logical or with horizontal dash | infix | 265 | 4 | 4 | |
⩞ | ⩞ | logical and with double overbar | infix | 265 | 4 | 4 | |
⩟ | ⩟ | logical and with underbar | infix | 265 | 4 | 4 | |
⩠ | ⩠ | logical and with double underbar | infix | 265 | 4 | 4 | |
⩡ | ⩡ | small vee with underbar | infix | 265 | 4 | 4 | |
⩢ | ⩢ | logical or with double overbar | infix | 265 | 4 | 4 | |
⩣ | ⩣ | logical or with double underbar | infix | 265 | 4 | 4 | |
⩤ | ⩤ | z notation domain antirestriction | infix | 265 | 4 | 4 | |
⩥ | ⩥ | z notation range antirestriction | infix | 265 | 4 | 4 | |
⩦ | ⩦ | equals sign with dot below | infix | 265 | 5 | 5 | |
⩧ | ⩧ | identical with dot above | infix | 265 | 5 | 5 | |
⩨ | ⩨ | triple horizontal bar with double vertical stroke | infix | 265 | 5 | 5 | |
⩩ | ⩩ | triple horizontal bar with triple vertical stroke | infix | 265 | 5 | 5 | |
⩪ | ⩪ | tilde operator with dot above | infix | 265 | 5 | 5 | |
⩫ | ⩫ | tilde operator with rising dots | infix | 265 | 5 | 5 | |
⩬ | ⩬ | similar minus similar | infix | 265 | 5 | 5 | |
⩭ | ⩭ | congruent with dot above | infix | 265 | 5 | 5 | |
⩮ | ⩮ | equals with asterisk | infix | 265 | 5 | 5 | |
⩯ | ⩯ | almost equal to with circumflex accent | infix | 265 | 5 | 5 | |
⩰ | ⩰ | approximately equal or equal to | infix | 265 | 5 | 5 | |
⩱ | ⩱ | equals sign above plus sign | infix | 265 | 4 | 4 | |
⩲ | ⩲ | plus sign above equals sign | infix | 265 | 4 | 4 | |
⩳ | ⩳ | equals sign above tilde operator | infix | 265 | 5 | 5 | |
⩴ | ⩴ | double colon equal | infix | 265 | 5 | 5 | |
⩵ | ⩵ | two consecutive equals signs | infix | 265 | 5 | 5 | |
⩶ | ⩶ | three consecutive equals signs | infix | 265 | 5 | 5 | |
⩷ | ⩷ | equals sign with two dots above and two dots below | infix | 265 | 5 | 5 | |
⩸ | ⩸ | equivalent with four dots above | infix | 265 | 5 | 5 | |
⩹ | ⩹ | less-than with circle inside | infix | 265 | 5 | 5 | |
⩺ | ⩺ | greater-than with circle inside | infix | 265 | 5 | 5 | |
⩻ | ⩻ | less-than with question mark above | infix | 265 | 5 | 5 | |
⩼ | ⩼ | greater-than with question mark above | infix | 265 | 5 | 5 | |
⩽ | ⩽ | less-than or slanted equal to | infix | 265 | 5 | 5 | |
⩽̸ | ⩽̸ | less-than or slanted equal to with slash | infix | 265 | 5 | 5 | |
⩾ | ⩾ | greater-than or slanted equal to | infix | 265 | 5 | 5 | |
⩾̸ | ⩾̸ | greater-than or slanted equal to with slash | infix | 265 | 5 | 5 | |
⩿ | ⩿ | less-than or slanted equal to with dot inside | infix | 265 | 5 | 5 | |
⪀ | ⪀ | greater-than or slanted equal to with dot inside | infix | 265 | 5 | 5 | |
⪁ | ⪁ | less-than or slanted equal to with dot above | infix | 265 | 5 | 5 | |
⪂ | ⪂ | greater-than or slanted equal to with dot above | infix | 265 | 5 | 5 | |
⪃ | ⪃ | less-than or slanted equal to with dot above right | infix | 265 | 5 | 5 | |
⪄ | ⪄ | greater-than or slanted equal to with dot above left | infix | 265 | 5 | 5 | |
⪅ | ⪅ | less-than or approximate | infix | 265 | 5 | 5 | |
⪆ | ⪆ | greater-than or approximate | infix | 265 | 5 | 5 | |
⪉ | ⪉ | less-than and not approximate | infix | 265 | 5 | 5 | |
⪊ | ⪊ | greater-than and not approximate | infix | 265 | 5 | 5 | |
⪋ | ⪋ | less-than above double-line equal above greater-than | infix | 265 | 5 | 5 | |
⪌ | ⪌ | greater-than above double-line equal above less-than | infix | 265 | 5 | 5 | |
⪍ | ⪍ | less-than above similar or equal | infix | 265 | 5 | 5 | |
⪎ | ⪎ | greater-than above similar or equal | infix | 265 | 5 | 5 | |
⪏ | ⪏ | less-than above similar above greater-than | infix | 265 | 5 | 5 | |
⪐ | ⪐ | greater-than above similar above less-than | infix | 265 | 5 | 5 | |
⪑ | ⪑ | less-than above greater-than above double-line equal | infix | 265 | 5 | 5 | |
⪒ | ⪒ | greater-than above less-than above double-line equal | infix | 265 | 5 | 5 | |
⪓ | ⪓ | less-than above slanted equal above greater-than above slanted equal | infix | 265 | 5 | 5 | |
⪔ | ⪔ | greater-than above slanted equal above less-than above slanted equal | infix | 265 | 5 | 5 | |
⪕ | ⪕ | slanted equal to or less-than | infix | 265 | 5 | 5 | |
⪖ | ⪖ | slanted equal to or greater-than | infix | 265 | 5 | 5 | |
⪗ | ⪗ | slanted equal to or less-than with dot inside | infix | 265 | 5 | 5 | |
⪘ | ⪘ | slanted equal to or greater-than with dot inside | infix | 265 | 5 | 5 | |
⪙ | ⪙ | double-line equal to or less-than | infix | 265 | 5 | 5 | |
⪚ | ⪚ | double-line equal to or greater-than | infix | 265 | 5 | 5 | |
⪛ | ⪛ | double-line slanted equal to or less-than | infix | 265 | 5 | 5 | |
⪜ | ⪜ | double-line slanted equal to or greater-than | infix | 265 | 5 | 5 | |
⪝ | ⪝ | similar or less-than | infix | 265 | 5 | 5 | |
⪞ | ⪞ | similar or greater-than | infix | 265 | 5 | 5 | |
⪟ | ⪟ | similar above less-than above equals sign | infix | 265 | 5 | 5 | |
⪠ | ⪠ | similar above greater-than above equals sign | infix | 265 | 5 | 5 | |
⪡ | ⪡ | double nested less-than | infix | 265 | 5 | 5 | |
⪡̸ | ⪡̸ | double nested less-than with slash | infix | 265 | 5 | 5 | |
⪢ | ⪢ | double nested greater-than | infix | 265 | 5 | 5 | |
⪢̸ | ⪢̸ | double nested greater-than with slash | infix | 265 | 5 | 5 | |
⪣ | ⪣ | double nested less-than with underbar | infix | 265 | 5 | 5 | |
⪤ | ⪤ | greater-than overlapping less-than | infix | 265 | 5 | 5 | |
⪥ | ⪥ | greater-than beside less-than | infix | 265 | 5 | 5 | |
⪦ | ⪦ | less-than closed by curve | infix | 265 | 5 | 5 | |
⪧ | ⪧ | greater-than closed by curve | infix | 265 | 5 | 5 | |
⪨ | ⪨ | less-than closed by curve above slanted equal | infix | 265 | 5 | 5 | |
⪩ | ⪩ | greater-than closed by curve above slanted equal | infix | 265 | 5 | 5 | |
⪪ | ⪪ | smaller than | infix | 265 | 5 | 5 | |
⪫ | ⪫ | larger than | infix | 265 | 5 | 5 | |
⪬ | ⪬ | smaller than or equal to | infix | 265 | 5 | 5 | |
⪭ | ⪭ | larger than or equal to | infix | 265 | 5 | 5 | |
⪮ | ⪮ | equals sign with bumpy above | infix | 265 | 5 | 5 | |
⪱ | ⪱ | precedes above single-line not equal to | infix | 265 | 5 | 5 | |
⪲ | ⪲ | succeeds above single-line not equal to | infix | 265 | 5 | 5 | |
⪳ | ⪳ | precedes above equals sign | infix | 265 | 5 | 5 | |
⪴ | ⪴ | succeeds above equals sign | infix | 265 | 5 | 5 | |
⪵ | ⪵ | precedes above not equal to | infix | 265 | 5 | 5 | |
⪶ | ⪶ | succeeds above not equal to | infix | 265 | 5 | 5 | |
⪷ | ⪷ | precedes above almost equal to | infix | 265 | 5 | 5 | |
⪸ | ⪸ | succeeds above almost equal to | infix | 265 | 5 | 5 | |
⪹ | ⪹ | precedes above not almost equal to | infix | 265 | 5 | 5 | |
⪺ | ⪺ | succeeds above not almost equal to | infix | 265 | 5 | 5 | |
⪻ | ⪻ | double precedes | infix | 265 | 5 | 5 | |
⪼ | ⪼ | double succeeds | infix | 265 | 5 | 5 | |
⪽ | ⪽ | subset with dot | infix | 265 | 5 | 5 | |
⪾ | ⪾ | superset with dot | infix | 265 | 5 | 5 | |
⪿ | ⪿ | subset with plus sign below | infix | 265 | 5 | 5 | |
⫀ | ⫀ | superset with plus sign below | infix | 265 | 5 | 5 | |
⫁ | ⫁ | subset with multiplication sign below | infix | 265 | 5 | 5 | |
⫂ | ⫂ | superset with multiplication sign below | infix | 265 | 5 | 5 | |
⫃ | ⫃ | subset of or equal to with dot above | infix | 265 | 5 | 5 | |
⫄ | ⫄ | superset of or equal to with dot above | infix | 265 | 5 | 5 | |
⫅ | ⫅ | subset of above equals sign | infix | 265 | 5 | 5 | |
⫆ | ⫆ | superset of above equals sign | infix | 265 | 5 | 5 | |
⫇ | ⫇ | subset of above tilde operator | infix | 265 | 5 | 5 | |
⫈ | ⫈ | superset of above tilde operator | infix | 265 | 5 | 5 | |
⫉ | ⫉ | subset of above almost equal to | infix | 265 | 5 | 5 | |
⫊ | ⫊ | superset of above almost equal to | infix | 265 | 5 | 5 | |
⫋ | ⫋ | subset of above not equal to | infix | 265 | 5 | 5 | |
⫌ | ⫌ | superset of above not equal to | infix | 265 | 5 | 5 | |
⫍ | ⫍ | square left open box operator | infix | 265 | 5 | 5 | |
⫎ | ⫎ | square right open box operator | infix | 265 | 5 | 5 | |
⫏ | ⫏ | closed subset | infix | 265 | 5 | 5 | |
⫐ | ⫐ | closed superset | infix | 265 | 5 | 5 | |
⫑ | ⫑ | closed subset or equal to | infix | 265 | 5 | 5 | |
⫒ | ⫒ | closed superset or equal to | infix | 265 | 5 | 5 | |
⫓ | ⫓ | subset above superset | infix | 265 | 5 | 5 | |
⫔ | ⫔ | superset above subset | infix | 265 | 5 | 5 | |
⫕ | ⫕ | subset above subset | infix | 265 | 5 | 5 | |
⫖ | ⫖ | superset above superset | infix | 265 | 5 | 5 | |
⫗ | ⫗ | superset beside subset | infix | 265 | 5 | 5 | |
⫘ | ⫘ | superset beside and joined by dash with subset | infix | 265 | 5 | 5 | |
⫙ | ⫙ | element of opening downwards | infix | 265 | 5 | 5 | |
⫚ | ⫚ | pitchfork with tee top | infix | 265 | 5 | 5 | |
⫛ | ⫛ | transversal intersection | infix | 265 | 5 | 5 | |
⫝ | ⫝ | nonforking | infix | 265 | 5 | 5 | |
⫝̸ | ⫝̸ | nonforking with slash | infix | 265 | 5 | 5 | |
⫞ | ⫞ | short left tack | infix | 265 | 5 | 5 | |
⫟ | ⫟ | short down tack | infix | 265 | 5 | 5 | |
⫠ | ⫠ | short up tack | infix | 265 | 5 | 5 | |
⫡ | ⫡ | perpendicular with s | infix | 265 | 5 | 5 | |
⫢ | ⫢ | vertical bar triple right turnstile | infix | 265 | 5 | 5 | |
⫣ | ⫣ | double vertical bar left turnstile | infix | 265 | 5 | 5 | |
⫤ | ⫤ | vertical bar double left turnstile | infix | 265 | 5 | 5 | |
⫥ | ⫥ | double vertical bar double left turnstile | infix | 265 | 5 | 5 | |
⫦ | ⫦ | long dash from left member of double vertical | infix | 265 | 5 | 5 | |
⫧ | ⫧ | short down tack with overbar | infix | 265 | 5 | 5 | |
⫨ | ⫨ | short up tack with underbar | infix | 265 | 5 | 5 | |
⫩ | ⫩ | short up tack above short down tack | infix | 265 | 5 | 5 | |
⫪ | ⫪ | double down tack | infix | 265 | 5 | 5 | |
⫫ | ⫫ | double up tack | infix | 265 | 5 | 5 | |
⫬ | ⫬ | double stroke not sign | infix | 265 | 5 | 5 | |
⫭ | ⫭ | reversed double stroke not sign | infix | 265 | 5 | 5 | |
⫮ | ⫮ | does not divide with reversed negation slash | infix | 265 | 5 | 5 | |
⫯ | ⫯ | vertical line with circle above | infix | 265 | 5 | 5 | |
⫰ | ⫰ | vertical line with circle below | infix | 265 | 5 | 5 | |
⫱ | ⫱ | down tack with circle below | infix | 265 | 5 | 5 | |
⫲ | ⫲ | parallel with horizontal stroke | infix | 265 | 5 | 5 | |
⫳ | ⫳ | parallel with tilde operator | infix | 265 | 5 | 5 | |
⫴ | ⫴ | triple vertical bar binary relation | infix | 265 | 4 | 4 | |
⫵ | ⫵ | triple vertical bar with horizontal stroke | infix | 265 | 4 | 4 | |
⫶ | ⫶ | triple colon operator | infix | 265 | 4 | 4 | |
⫷ | ⫷ | triple nested less-than | infix | 265 | 5 | 5 | |
⫸ | ⫸ | triple nested greater-than | infix | 265 | 5 | 5 | |
⫹ | ⫹ | double-line slanted less-than or equal to | infix | 265 | 5 | 5 | |
⫺ | ⫺ | double-line slanted greater-than or equal to | infix | 265 | 5 | 5 | |
⫻ | ⫻ | triple solidus binary relation | infix | 265 | 4 | 4 | |
⫽ | ⫽ | double solidus operator | infix | 265 | 4 | 4 | |
⫾ | ⫾ | white vertical bar | infix | 265 | 3 | 3 | |
| | | | vertical line | infix | 270 | 2 | 2 | fence, stretchy, symmetric |
|| | || | multiple character operator: || | infix | 270 | 2 | 2 | fence, stretchy, symmetric |
||| | ||| | multiple character operator: ||| | infix | 270 | 2 | 2 | fence, stretchy, symmetric |
← | ← | leftwards arrow | infix | 270 | 5 | 5 | stretchy, accent |
↑ | ↑ | upwards arrow | infix | 270 | 5 | 5 | stretchy |
→ | → | rightwards arrow | infix | 270 | 5 | 5 | stretchy, accent |
↓ | ↓ | downwards arrow | infix | 270 | 5 | 5 | stretchy |
↔ | ↔ | left right arrow | infix | 270 | 5 | 5 | stretchy, accent |
↕ | ↕ | up down arrow | infix | 270 | 5 | 5 | stretchy |
↖ | ↖ | north west arrow | infix | 270 | 5 | 5 | stretchy |
↗ | ↗ | north east arrow | infix | 270 | 5 | 5 | stretchy |
↘ | ↘ | south east arrow | infix | 270 | 5 | 5 | stretchy |
↙ | ↙ | south west arrow | infix | 270 | 5 | 5 | stretchy |
↚ | ↚ | leftwards arrow with stroke | infix | 270 | 5 | 5 | accent |
↛ | ↛ | rightwards arrow with stroke | infix | 270 | 5 | 5 | accent |
↜ | ↜ | leftwards wave arrow | infix | 270 | 5 | 5 | stretchy, accent |
↝ | ↝ | rightwards wave arrow | infix | 270 | 5 | 5 | stretchy, accent |
↞ | ↞ | leftwards two headed arrow | infix | 270 | 5 | 5 | stretchy, accent |
↟ | ↟ | upwards two headed arrow | infix | 270 | 5 | 5 | stretchy, accent |
↠ | ↠ | rightwards two headed arrow | infix | 270 | 5 | 5 | stretchy, accent |
↡ | ↡ | downwards two headed arrow | infix | 270 | 5 | 5 | stretchy |
↢ | ↢ | leftwards arrow with tail | infix | 270 | 5 | 5 | stretchy, accent |
↣ | ↣ | rightwards arrow with tail | infix | 270 | 5 | 5 | stretchy, accent |
↤ | ↤ | leftwards arrow from bar | infix | 270 | 5 | 5 | stretchy, accent |
↥ | ↥ | upwards arrow from bar | infix | 270 | 5 | 5 | stretchy |
↦ | ↦ | rightwards arrow from bar | infix | 270 | 5 | 5 | stretchy, accent |
↧ | ↧ | downwards arrow from bar | infix | 270 | 5 | 5 | stretchy |
↨ | ↨ | up down arrow with base | infix | 270 | 5 | 5 | stretchy |
↩ | ↩ | leftwards arrow with hook | infix | 270 | 5 | 5 | stretchy, accent |
↪ | ↪ | rightwards arrow with hook | infix | 270 | 5 | 5 | stretchy, accent |
↫ | ↫ | leftwards arrow with loop | infix | 270 | 5 | 5 | stretchy, accent |
↬ | ↬ | rightwards arrow with loop | infix | 270 | 5 | 5 | stretchy, accent |
↭ | ↭ | left right wave arrow | infix | 270 | 5 | 5 | stretchy, accent |
↮ | ↮ | left right arrow with stroke | infix | 270 | 5 | 5 | accent |
↯ | ↯ | downwards zigzag arrow | infix | 270 | 5 | 5 | stretchy |
↰ | ↰ | upwards arrow with tip leftwards | infix | 270 | 5 | 5 | stretchy |
↱ | ↱ | upwards arrow with tip rightwards | infix | 270 | 5 | 5 | stretchy |
↲ | ↲ | downwards arrow with tip leftwards | infix | 270 | 5 | 5 | stretchy |
↳ | ↳ | downwards arrow with tip rightwards | infix | 270 | 5 | 5 | stretchy |
↴ | ↴ | rightwards arrow with corner downwards | infix | 270 | 5 | 5 | stretchy |
↵ | ↵ | downwards arrow with corner leftwards | infix | 270 | 5 | 5 | stretchy |
↶ | ↶ | anticlockwise top semicircle arrow | infix | 270 | 5 | 5 | accent |
↷ | ↷ | clockwise top semicircle arrow | infix | 270 | 5 | 5 | accent |
↸ | ↸ | north west arrow to long bar | infix | 270 | 5 | 5 | |
↹ | ↹ | leftwards arrow to bar over rightwards arrow to bar | infix | 270 | 5 | 5 | stretchy, accent |
↺ | ↺ | anticlockwise open circle arrow | infix | 270 | 5 | 5 | |
↻ | ↻ | clockwise open circle arrow | infix | 270 | 5 | 5 | |
↼ | ↼ | leftwards harpoon with barb upwards | infix | 270 | 5 | 5 | stretchy, accent |
↽ | ↽ | leftwards harpoon with barb downwards | infix | 270 | 5 | 5 | stretchy, accent |
↾ | ↾ | upwards harpoon with barb rightwards | infix | 270 | 5 | 5 | stretchy |
↿ | ↿ | upwards harpoon with barb leftwards | infix | 270 | 5 | 5 | stretchy |
⇀ | ⇀ | rightwards harpoon with barb upwards | infix | 270 | 5 | 5 | stretchy, accent |
⇁ | ⇁ | rightwards harpoon with barb downwards | infix | 270 | 5 | 5 | stretchy, accent |
⇂ | ⇂ | downwards harpoon with barb rightwards | infix | 270 | 5 | 5 | stretchy |
⇃ | ⇃ | downwards harpoon with barb leftwards | infix | 270 | 5 | 5 | stretchy |
⇄ | ⇄ | rightwards arrow over leftwards arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇅ | ⇅ | upwards arrow leftwards of downwards arrow | infix | 270 | 5 | 5 | stretchy |
⇆ | ⇆ | leftwards arrow over rightwards arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇇ | ⇇ | leftwards paired arrows | infix | 270 | 5 | 5 | stretchy, accent |
⇈ | ⇈ | upwards paired arrows | infix | 270 | 5 | 5 | stretchy |
⇉ | ⇉ | rightwards paired arrows | infix | 270 | 5 | 5 | stretchy, accent |
⇊ | ⇊ | downwards paired arrows | infix | 270 | 5 | 5 | stretchy |
⇋ | ⇋ | leftwards harpoon over rightwards harpoon | infix | 270 | 5 | 5 | stretchy, accent |
⇌ | ⇌ | rightwards harpoon over leftwards harpoon | infix | 270 | 5 | 5 | stretchy, accent |
⇍ | ⇍ | leftwards double arrow with stroke | infix | 270 | 5 | 5 | accent |
⇎ | ⇎ | left right double arrow with stroke | infix | 270 | 5 | 5 | accent |
⇏ | ⇏ | rightwards double arrow with stroke | infix | 270 | 5 | 5 | accent |
⇐ | ⇐ | leftwards double arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇑ | ⇑ | upwards double arrow | infix | 270 | 5 | 5 | stretchy |
⇒ | ⇒ | rightwards double arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇓ | ⇓ | downwards double arrow | infix | 270 | 5 | 5 | stretchy |
⇔ | ⇔ | left right double arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇕ | ⇕ | up down double arrow | infix | 270 | 5 | 5 | stretchy |
⇖ | ⇖ | north west double arrow | infix | 270 | 5 | 5 | stretchy |
⇗ | ⇗ | north east double arrow | infix | 270 | 5 | 5 | stretchy |
⇘ | ⇘ | south east double arrow | infix | 270 | 5 | 5 | stretchy |
⇙ | ⇙ | south west double arrow | infix | 270 | 5 | 5 | stretchy |
⇚ | ⇚ | leftwards triple arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇛ | ⇛ | rightwards triple arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇜ | ⇜ | leftwards squiggle arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇝ | ⇝ | rightwards squiggle arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇞ | ⇞ | upwards arrow with double stroke | infix | 270 | 5 | 5 | |
⇟ | ⇟ | downwards arrow with double stroke | infix | 270 | 5 | 5 | |
⇠ | ⇠ | leftwards dashed arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇡ | ⇡ | upwards dashed arrow | infix | 270 | 5 | 5 | stretchy |
⇢ | ⇢ | rightwards dashed arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇣ | ⇣ | downwards dashed arrow | infix | 270 | 5 | 5 | stretchy |
⇤ | ⇤ | leftwards arrow to bar | infix | 270 | 5 | 5 | stretchy, accent |
⇥ | ⇥ | rightwards arrow to bar | infix | 270 | 5 | 5 | stretchy, accent |
⇦ | ⇦ | leftwards white arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇧ | ⇧ | upwards white arrow | infix | 270 | 5 | 5 | stretchy |
⇨ | ⇨ | rightwards white arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇩ | ⇩ | downwards white arrow | infix | 270 | 5 | 5 | stretchy |
⇪ | ⇪ | upwards white arrow from bar | infix | 270 | 5 | 5 | stretchy |
⇫ | ⇫ | upwards white arrow on pedestal | infix | 270 | 5 | 5 | stretchy |
⇬ | ⇬ | upwards white arrow on pedestal with horizontal bar | infix | 270 | 5 | 5 | stretchy |
⇭ | ⇭ | upwards white arrow on pedestal with vertical bar | infix | 270 | 5 | 5 | stretchy |
⇮ | ⇮ | upwards white double arrow | infix | 270 | 5 | 5 | stretchy |
⇯ | ⇯ | upwards white double arrow on pedestal | infix | 270 | 5 | 5 | stretchy |
⇰ | ⇰ | rightwards white arrow from wall | infix | 270 | 5 | 5 | stretchy, accent |
⇱ | ⇱ | north west arrow to corner | infix | 270 | 5 | 5 | |
⇲ | ⇲ | south east arrow to corner | infix | 270 | 5 | 5 | |
⇳ | ⇳ | up down white arrow | infix | 270 | 5 | 5 | stretchy |
⇴ | ⇴ | right arrow with small circle | infix | 270 | 5 | 5 | accent |
⇵ | ⇵ | downwards arrow leftwards of upwards arrow | infix | 270 | 5 | 5 | stretchy |
⇶ | ⇶ | three rightwards arrows | infix | 270 | 5 | 5 | stretchy, accent |
⇷ | ⇷ | leftwards arrow with vertical stroke | infix | 270 | 5 | 5 | accent |
⇸ | ⇸ | rightwards arrow with vertical stroke | infix | 270 | 5 | 5 | accent |
⇹ | ⇹ | left right arrow with vertical stroke | infix | 270 | 5 | 5 | accent |
⇺ | ⇺ | leftwards arrow with double vertical stroke | infix | 270 | 5 | 5 | accent |
⇻ | ⇻ | rightwards arrow with double vertical stroke | infix | 270 | 5 | 5 | accent |
⇼ | ⇼ | left right arrow with double vertical stroke | infix | 270 | 5 | 5 | accent |
⇽ | ⇽ | leftwards open-headed arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇾ | ⇾ | rightwards open-headed arrow | infix | 270 | 5 | 5 | stretchy, accent |
⇿ | ⇿ | left right open-headed arrow | infix | 270 | 5 | 5 | stretchy, accent |
⊸ | ⊸ | multimap | infix | 270 | 5 | 5 | |
⟰ | ⟰ | upwards quadruple arrow | infix | 270 | 5 | 5 | stretchy |
⟱ | ⟱ | downwards quadruple arrow | infix | 270 | 5 | 5 | stretchy |
⟵ | ⟵ | long leftwards arrow | infix | 270 | 5 | 5 | stretchy, accent |
⟶ | ⟶ | long rightwards arrow | infix | 270 | 5 | 5 | stretchy, accent |
⟷ | ⟷ | long left right arrow | infix | 270 | 5 | 5 | stretchy, accent |
⟸ | ⟸ | long leftwards double arrow | infix | 270 | 5 | 5 | stretchy, accent |
⟹ | ⟹ | long rightwards double arrow | infix | 270 | 5 | 5 | stretchy, accent |
⟺ | ⟺ | long left right double arrow | infix | 270 | 5 | 5 | stretchy, accent |
⟻ | ⟻ | long leftwards arrow from bar | infix | 270 | 5 | 5 | stretchy, accent |
⟼ | ⟼ | long rightwards arrow from bar | infix | 270 | 5 | 5 | stretchy, accent |
⟽ | ⟽ | long leftwards double arrow from bar | infix | 270 | 5 | 5 | stretchy, accent |
⟾ | ⟾ | long rightwards double arrow from bar | infix | 270 | 5 | 5 | stretchy, accent |
⟿ | ⟿ | long rightwards squiggle arrow | infix | 270 | 5 | 5 | stretchy, accent |
⤀ | ⤀ | rightwards two-headed arrow with vertical stroke | infix | 270 | 5 | 5 | accent |
⤁ | ⤁ | rightwards two-headed arrow with double vertical stroke | infix | 270 | 5 | 5 | accent |
⤂ | ⤂ | leftwards double arrow with vertical stroke | infix | 270 | 5 | 5 | accent |
⤃ | ⤃ | rightwards double arrow with vertical stroke | infix | 270 | 5 | 5 | accent |
⤄ | ⤄ | left right double arrow with vertical stroke | infix | 270 | 5 | 5 | accent |
⤅ | ⤅ | rightwards two-headed arrow from bar | infix | 270 | 5 | 5 | accent |
⤆ | ⤆ | leftwards double arrow from bar | infix | 270 | 5 | 5 | accent |
⤇ | ⤇ | rightwards double arrow from bar | infix | 270 | 5 | 5 | accent |
⤈ | ⤈ | downwards arrow with horizontal stroke | infix | 270 | 5 | 5 | |
⤉ | ⤉ | upwards arrow with horizontal stroke | infix | 270 | 5 | 5 | |
⤊ | ⤊ | upwards triple arrow | infix | 270 | 5 | 5 | stretchy |
⤋ | ⤋ | downwards triple arrow | infix | 270 | 5 | 5 | stretchy |
⤌ | ⤌ | leftwards double dash arrow | infix | 270 | 5 | 5 | stretchy, accent |
⤍ | ⤍ | rightwards double dash arrow | infix | 270 | 5 | 5 | stretchy, accent |
⤎ | ⤎ | leftwards triple dash arrow | infix | 270 | 5 | 5 | stretchy, accent |
⤏ | ⤏ | rightwards triple dash arrow | infix | 270 | 5 | 5 | stretchy, accent |
⤐ | ⤐ | rightwards two-headed triple dash arrow | infix | 270 | 5 | 5 | stretchy, accent |
⤑ | ⤑ | rightwards arrow with dotted stem | infix | 270 | 5 | 5 | accent |
⤒ | ⤒ | upwards arrow to bar | infix | 270 | 5 | 5 | stretchy |
⤓ | ⤓ | downwards arrow to bar | infix | 270 | 5 | 5 | stretchy |
⤔ | ⤔ | rightwards arrow with tail with vertical stroke | infix | 270 | 5 | 5 | accent |
⤕ | ⤕ | rightwards arrow with tail with double vertical stroke | infix | 270 | 5 | 5 | accent |
⤖ | ⤖ | rightwards two-headed arrow with tail | infix | 270 | 5 | 5 | accent |
⤗ | ⤗ | rightwards two-headed arrow with tail with vertical stroke | infix | 270 | 5 | 5 | accent |
⤘ | ⤘ | rightwards two-headed arrow with tail with double vertical stroke | infix | 270 | 5 | 5 | accent |
⤙ | ⤙ | leftwards arrow-tail | infix | 270 | 5 | 5 | accent |
⤚ | ⤚ | rightwards arrow-tail | infix | 270 | 5 | 5 | accent |
⤛ | ⤛ | leftwards double arrow-tail | infix | 270 | 5 | 5 | accent |
⤜ | ⤜ | rightwards double arrow-tail | infix | 270 | 5 | 5 | accent |
⤝ | ⤝ | leftwards arrow to black diamond | infix | 270 | 5 | 5 | accent |
⤞ | ⤞ | rightwards arrow to black diamond | infix | 270 | 5 | 5 | accent |
⤟ | ⤟ | leftwards arrow from bar to black diamond | infix | 270 | 5 | 5 | accent |
⤠ | ⤠ | rightwards arrow from bar to black diamond | infix | 270 | 5 | 5 | accent |
⤡ | ⤡ | north west and south east arrow | infix | 270 | 5 | 5 | stretchy |
⤢ | ⤢ | north east and south west arrow | infix | 270 | 5 | 5 | stretchy |
⤣ | ⤣ | north west arrow with hook | infix | 270 | 5 | 5 | |
⤤ | ⤤ | north east arrow with hook | infix | 270 | 5 | 5 | |
⤥ | ⤥ | south east arrow with hook | infix | 270 | 5 | 5 | |
⤦ | ⤦ | south west arrow with hook | infix | 270 | 5 | 5 | |
⤧ | ⤧ | north west arrow and north east arrow | infix | 270 | 5 | 5 | |
⤨ | ⤨ | north east arrow and south east arrow | infix | 270 | 5 | 5 | |
⤩ | ⤩ | south east arrow and south west arrow | infix | 270 | 5 | 5 | |
⤪ | ⤪ | south west arrow and north west arrow | infix | 270 | 5 | 5 | |
⤫ | ⤫ | rising diagonal crossing falling diagonal | infix | 270 | 5 | 5 | |
⤬ | ⤬ | falling diagonal crossing rising diagonal | infix | 270 | 5 | 5 | |
⤭ | ⤭ | south east arrow crossing north east arrow | infix | 270 | 5 | 5 | |
⤮ | ⤮ | north east arrow crossing south east arrow | infix | 270 | 5 | 5 | |
⤯ | ⤯ | falling diagonal crossing north east arrow | infix | 270 | 5 | 5 | |
⤰ | ⤰ | rising diagonal crossing south east arrow | infix | 270 | 5 | 5 | |
⤱ | ⤱ | north east arrow crossing north west arrow | infix | 270 | 5 | 5 | |
⤲ | ⤲ | north west arrow crossing north east arrow | infix | 270 | 5 | 5 | |
⤳ | ⤳ | wave arrow pointing directly right | infix | 270 | 5 | 5 | accent |
⤴ | ⤴ | arrow pointing rightwards then curving upwards | infix | 270 | 5 | 5 | |
⤵ | ⤵ | arrow pointing rightwards then curving downwards | infix | 270 | 5 | 5 | |
⤶ | ⤶ | arrow pointing downwards then curving leftwards | infix | 270 | 5 | 5 | |
⤷ | ⤷ | arrow pointing downwards then curving rightwards | infix | 270 | 5 | 5 | |
⤸ | ⤸ | right-side arc clockwise arrow | infix | 270 | 5 | 5 | |
⤹ | ⤹ | left-side arc anticlockwise arrow | infix | 270 | 5 | 5 | |
⤺ | ⤺ | top arc anticlockwise arrow | infix | 270 | 5 | 5 | accent |
⤻ | ⤻ | bottom arc anticlockwise arrow | infix | 270 | 5 | 5 | accent |
⤼ | ⤼ | top arc clockwise arrow with minus | infix | 270 | 5 | 5 | accent |
⤽ | ⤽ | top arc anticlockwise arrow with plus | infix | 270 | 5 | 5 | accent |
⤾ | ⤾ | lower right semicircular clockwise arrow | infix | 270 | 5 | 5 | |
⤿ | ⤿ | lower left semicircular anticlockwise arrow | infix | 270 | 5 | 5 | |
⥀ | ⥀ | anticlockwise closed circle arrow | infix | 270 | 5 | 5 | |
⥁ | ⥁ | clockwise closed circle arrow | infix | 270 | 5 | 5 | |
⥂ | ⥂ | rightwards arrow above short leftwards arrow | infix | 270 | 5 | 5 | accent |
⥃ | ⥃ | leftwards arrow above short rightwards arrow | infix | 270 | 5 | 5 | accent |
⥄ | ⥄ | short rightwards arrow above leftwards arrow | infix | 270 | 5 | 5 | accent |
⥅ | ⥅ | rightwards arrow with plus below | infix | 270 | 5 | 5 | accent |
⥆ | ⥆ | leftwards arrow with plus below | infix | 270 | 5 | 5 | accent |
⥇ | ⥇ | rightwards arrow through x | infix | 270 | 5 | 5 | accent |
⥈ | ⥈ | left right arrow through small circle | infix | 270 | 5 | 5 | accent |
⥉ | ⥉ | upwards two-headed arrow from small circle | infix | 270 | 5 | 5 | |
⥊ | ⥊ | left barb up right barb down harpoon | infix | 270 | 5 | 5 | accent |
⥋ | ⥋ | left barb down right barb up harpoon | infix | 270 | 5 | 5 | accent |
⥌ | ⥌ | up barb right down barb left harpoon | infix | 270 | 5 | 5 | |
⥍ | ⥍ | up barb left down barb right harpoon | infix | 270 | 5 | 5 | |
⥎ | ⥎ | left barb up right barb up harpoon | infix | 270 | 5 | 5 | stretchy, accent |
⥏ | ⥏ | up barb right down barb right harpoon | infix | 270 | 5 | 5 | stretchy |
⥐ | ⥐ | left barb down right barb down harpoon | infix | 270 | 5 | 5 | stretchy, accent |
⥑ | ⥑ | up barb left down barb left harpoon | infix | 270 | 5 | 5 | stretchy |
⥒ | ⥒ | leftwards harpoon with barb up to bar | infix | 270 | 5 | 5 | stretchy, accent |
⥓ | ⥓ | rightwards harpoon with barb up to bar | infix | 270 | 5 | 5 | stretchy, accent |
⥔ | ⥔ | upwards harpoon with barb right to bar | infix | 270 | 5 | 5 | stretchy |
⥕ | ⥕ | downwards harpoon with barb right to bar | infix | 270 | 5 | 5 | stretchy |
⥖ | ⥖ | leftwards harpoon with barb down to bar | infix | 270 | 5 | 5 | stretchy |
⥗ | ⥗ | rightwards harpoon with barb down to bar | infix | 270 | 5 | 5 | stretchy |
⥘ | ⥘ | upwards harpoon with barb left to bar | infix | 270 | 5 | 5 | stretchy |
⥙ | ⥙ | downwards harpoon with barb left to bar | infix | 270 | 5 | 5 | stretchy |
⥚ | ⥚ | leftwards harpoon with barb up from bar | infix | 270 | 5 | 5 | stretchy, accent |
⥛ | ⥛ | rightwards harpoon with barb up from bar | infix | 270 | 5 | 5 | stretchy, accent |
⥜ | ⥜ | upwards harpoon with barb right from bar | infix | 270 | 5 | 5 | stretchy |
⥝ | ⥝ | downwards harpoon with barb right from bar | infix | 270 | 5 | 5 | stretchy |
⥞ | ⥞ | leftwards harpoon with barb down from bar | infix | 270 | 5 | 5 | stretchy, accent |
⥟ | ⥟ | rightwards harpoon with barb down from bar | infix | 270 | 5 | 5 | stretchy, accent |
⥠ | ⥠ | upwards harpoon with barb left from bar | infix | 270 | 5 | 5 | stretchy |
⥡ | ⥡ | downwards harpoon with barb left from bar | infix | 270 | 5 | 5 | stretchy |
⥢ | ⥢ | leftwards harpoon with barb up above leftwards harpoon with barb down | infix | 270 | 5 | 5 | accent |
⥣ | ⥣ | upwards harpoon with barb left beside upwards harpoon with barb right | infix | 270 | 5 | 5 | |
⥤ | ⥤ | rightwards harpoon with barb up above rightwards harpoon with barb down | infix | 270 | 5 | 5 | accent |
⥥ | ⥥ | downwards harpoon with barb left beside downwards harpoon with barb right | infix | 270 | 5 | 5 | |
⥦ | ⥦ | leftwards harpoon with barb up above rightwards harpoon with barb up | infix | 270 | 5 | 5 | accent |
⥧ | ⥧ | leftwards harpoon with barb down above rightwards harpoon with barb down | infix | 270 | 5 | 5 | accent |
⥨ | ⥨ | rightwards harpoon with barb up above leftwards harpoon with barb up | infix | 270 | 5 | 5 | accent |
⥩ | ⥩ | rightwards harpoon with barb down above leftwards harpoon with barb down | infix | 270 | 5 | 5 | accent |
⥪ | ⥪ | leftwards harpoon with barb up above long dash | infix | 270 | 5 | 5 | accent |
⥫ | ⥫ | leftwards harpoon with barb down below long dash | infix | 270 | 5 | 5 | accent |
⥬ | ⥬ | rightwards harpoon with barb up above long dash | infix | 270 | 5 | 5 | accent |
⥭ | ⥭ | rightwards harpoon with barb down below long dash | infix | 270 | 5 | 5 | accent |
⥮ | ⥮ | upwards harpoon with barb left beside downwards harpoon with barb right | infix | 270 | 5 | 5 | stretchy |
⥯ | ⥯ | downwards harpoon with barb left beside upwards harpoon with barb right | infix | 270 | 5 | 5 | stretchy |
⥰ | ⥰ | right double arrow with rounded head | infix | 270 | 5 | 5 | accent |
⥱ | ⥱ | equals sign above rightwards arrow | infix | 270 | 5 | 5 | accent |
⥲ | ⥲ | tilde operator above rightwards arrow | infix | 270 | 5 | 5 | accent |
⥳ | ⥳ | leftwards arrow above tilde operator | infix | 270 | 5 | 5 | accent |
⥴ | ⥴ | rightwards arrow above tilde operator | infix | 270 | 5 | 5 | accent |
⥵ | ⥵ | rightwards arrow above almost equal to | infix | 270 | 5 | 5 | accent |
⥶ | ⥶ | less-than above leftwards arrow | infix | 270 | 5 | 5 | accent |
⥷ | ⥷ | leftwards arrow through less-than | infix | 270 | 5 | 5 | accent |
⥸ | ⥸ | greater-than above rightwards arrow | infix | 270 | 5 | 5 | accent |
⥹ | ⥹ | subset above rightwards arrow | infix | 270 | 5 | 5 | accent |
⥺ | ⥺ | leftwards arrow through subset | infix | 270 | 5 | 5 | accent |
⥻ | ⥻ | superset above leftwards arrow | infix | 270 | 5 | 5 | accent |
⥼ | ⥼ | left fish tail | infix | 270 | 5 | 5 | accent |
⥽ | ⥽ | right fish tail | infix | 270 | 5 | 5 | accent |
⥾ | ⥾ | up fish tail | infix | 270 | 5 | 5 | |
⥿ | ⥿ | down fish tail | infix | 270 | 5 | 5 | |
⦙ | ⦙ | dotted fence | infix | 270 | 3 | 3 | |
⦚ | ⦚ | vertical zigzag line | infix | 270 | 3 | 3 | |
⦛ | ⦛ | measured angle opening left | infix | 270 | 3 | 3 | |
⦜ | ⦜ | right angle variant with square | infix | 270 | 3 | 3 | |
⦝ | ⦝ | measured right angle with dot | infix | 270 | 3 | 3 | |
⦞ | ⦞ | angle with s inside | infix | 270 | 3 | 3 | |
⦟ | ⦟ | acute angle | infix | 270 | 3 | 3 | |
⧟ | ⧟ | double-ended multimap | infix | 270 | 3 | 3 | |
⧯ | ⧯ | error-barred black square | infix | 270 | 3 | 3 | |
⧴ | ⧴ | rule-delayed | infix | 270 | 5 | 5 | |
⭅ | ⭅ | leftwards quadruple arrow | infix | 270 | 5 | 5 | stretchy |
⭆ | ⭆ | rightwards quadruple arrow | infix | 270 | 5 | 5 | stretchy |
+ | + | plus sign | infix | 275 | 4 | 4 | |
+ | + | plus sign | prefix | 275 | 0 | 1 | |
- | - | hyphen-minus | infix | 275 | 4 | 4 | |
- | - | hyphen-minus | prefix | 275 | 0 | 1 | |
± | ± | plus-minus sign | infix | 275 | 4 | 4 | |
± | ± | plus-minus sign | prefix | 275 | 0 | 1 | |
− | − | minus sign | infix | 275 | 4 | 4 | |
− | − | minus sign | prefix | 275 | 0 | 1 | |
∓ | ∓ | minus-or-plus sign | infix | 275 | 4 | 4 | |
∓ | ∓ | minus-or-plus sign | prefix | 275 | 0 | 1 | |
∔ | ∔ | dot plus | infix | 275 | 4 | 4 | |
⊞ | ⊞ | squared plus | infix | 275 | 4 | 4 | |
⊟ | ⊟ | squared minus | infix | 275 | 4 | 4 | |
∑ | ∑ | n-ary summation | prefix | 290 | 1 | 2 | largeop, movablelimits, symmetric |
⨊ | ⨊ | modulo two sum | prefix | 290 | 1 | 2 | largeop, movablelimits, symmetric |
⨋ | ⨋ | summation with integral | prefix | 290 | 1 | 2 | largeop, symmetric |
∬ | ∬ | double integral | prefix | 300 | 0 | 1 | largeop, symmetric |
∭ | ∭ | triple integral | prefix | 300 | 0 | 1 | largeop, symmetric |
⊕ | ⊕ | circled plus | infix | 300 | 4 | 4 | |
⊖ | ⊖ | circled minus | infix | 300 | 4 | 4 | |
⊘ | ⊘ | circled division slash | infix | 300 | 4 | 4 | |
⨁ | ⨁ | n-ary circled plus operator | prefix | 300 | 1 | 2 | largeop, movablelimits, symmetric |
∫ | ∫ | integral | prefix | 310 | 0 | 1 | largeop, symmetric |
∮ | ∮ | contour integral | prefix | 310 | 0 | 1 | largeop, symmetric |
∯ | ∯ | surface integral | prefix | 310 | 0 | 1 | largeop, symmetric |
∰ | ∰ | volume integral | prefix | 310 | 0 | 1 | largeop, symmetric |
∱ | ∱ | clockwise integral | prefix | 310 | 0 | 1 | largeop, symmetric |
∲ | ∲ | clockwise contour integral | prefix | 310 | 0 | 1 | largeop, symmetric |
∳ | ∳ | anticlockwise contour integral | prefix | 310 | 0 | 1 | largeop, symmetric |
⨌ | ⨌ | quadruple integral operator | prefix | 310 | 0 | 1 | largeop, symmetric |
⨍ | ⨍ | finite part integral | prefix | 310 | 1 | 2 | largeop, symmetric |
⨎ | ⨎ | integral with double stroke | prefix | 310 | 1 | 2 | largeop, symmetric |
⨏ | ⨏ | integral average with slash | prefix | 310 | 1 | 2 | largeop, symmetric |
⨐ | ⨐ | circulation function | prefix | 310 | 1 | 2 | largeop, movablelimits, symmetric |
⨑ | ⨑ | anticlockwise integration | prefix | 310 | 1 | 2 | largeop, movablelimits, symmetric |
⨒ | ⨒ | line integration with rectangular path around pole | prefix | 310 | 1 | 2 | largeop, movablelimits, symmetric |
⨓ | ⨓ | line integration with semicircular path around pole | prefix | 310 | 1 | 2 | largeop, movablelimits, symmetric |
⨔ | ⨔ | line integration not including the pole | prefix | 310 | 1 | 2 | largeop, movablelimits, symmetric |
⨕ | ⨕ | integral around a point operator | prefix | 310 | 1 | 2 | largeop, symmetric |
⨖ | ⨖ | quaternion integral operator | prefix | 310 | 1 | 2 | largeop, symmetric |
⨗ | ⨗ | integral with leftwards arrow with hook | prefix | 310 | 1 | 2 | largeop, symmetric |
⨘ | ⨘ | integral with times sign | prefix | 310 | 1 | 2 | largeop, symmetric |
⨙ | ⨙ | integral with intersection | prefix | 310 | 1 | 2 | largeop, symmetric |
⨚ | ⨚ | integral with union | prefix | 310 | 1 | 2 | largeop, symmetric |
⨛ | ⨛ | integral with overbar | prefix | 310 | 1 | 2 | largeop, symmetric |
⨜ | ⨜ | integral with underbar | prefix | 310 | 1 | 2 | largeop, symmetric |
⋃ | ⋃ | n-ary union | prefix | 320 | 1 | 2 | largeop, movablelimits, symmetric |
⨃ | ⨃ | n-ary union operator with dot | prefix | 320 | 1 | 2 | largeop, movablelimits, symmetric |
⨄ | ⨄ | n-ary union operator with plus | prefix | 320 | 1 | 2 | largeop, movablelimits, symmetric |
⋀ | ⋀ | n-ary logical and | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⋁ | ⋁ | n-ary logical or | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⋂ | ⋂ | n-ary intersection | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⨀ | ⨀ | n-ary circled dot operator | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⨂ | ⨂ | n-ary circled times operator | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⨅ | ⨅ | n-ary square intersection operator | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⨆ | ⨆ | n-ary square union operator | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⨇ | ⨇ | two logical and operator | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⨈ | ⨈ | two logical or operator | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⨉ | ⨉ | n-ary times operator | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⫼ | ⫼ | large triple vertical bar operator | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
⫿ | ⫿ | n-ary white vertical bar | prefix | 330 | 1 | 2 | largeop, movablelimits, symmetric |
≀ | ≀ | wreath product | infix | 340 | 4 | 4 | |
∏ | ∏ | n-ary product | prefix | 350 | 1 | 2 | largeop, movablelimits, symmetric |
∐ | ∐ | n-ary coproduct | prefix | 350 | 1 | 2 | largeop, movablelimits, symmetric |
∩ | ∩ | intersection | infix | 350 | 4 | 4 | |
∪ | ∪ | union | infix | 350 | 4 | 4 | |
* | * | asterisk | infix | 390 | 3 | 3 | |
. | . | full stop | infix | 390 | 3 | 3 | |
× | × | multiplication sign | infix | 390 | 4 | 4 | |
• | • | bullet | infix | 390 | 4 | 4 | |
⁃ | ⁃ | hyphen bullet | infix | 390 | 4 | 4 | |
⁢ | | invisible times | infix | 390 | 0 | 0 | |
⊠ | ⊠ | squared times | infix | 390 | 4 | 4 | |
⊡ | ⊡ | squared dot operator | infix | 390 | 4 | 4 | |
⋅ | ⋅ | dot operator | infix | 390 | 4 | 4 | |
⨯ | ⨯ | vector or cross product | infix | 390 | 4 | 4 | |
⨿ | ⨿ | amalgamation or coproduct | infix | 390 | 4 | 4 | |
· | · | middle dot | infix | 400 | 4 | 4 | |
⊗ | ⊗ | circled times | infix | 410 | 4 | 4 | |
% | % | percent sign | infix | 640 | 3 | 3 | |
\ | \ | reverse solidus | infix | 650 | 0 | 0 | |
∖ | ∖ | set minus | infix | 650 | 4 | 4 | |
/ | / | solidus | infix | 660 | 1 | 1 | |
÷ | ÷ | division sign | infix | 660 | 4 | 4 | |
∠ | ∠ | angle | prefix | 670 | 0 | 0 | |
∡ | ∡ | measured angle | prefix | 670 | 0 | 0 | |
∢ | ∢ | spherical angle | prefix | 670 | 0 | 0 | |
¬ | ¬ | not sign | prefix | 680 | 2 | 1 | |
⊙ | ⊙ | circled dot operator | infix | 710 | 4 | 4 | |
∂ | ∂ | partial differential | prefix | 740 | 2 | 1 | |
∇ | ∇ | nabla | prefix | 740 | 2 | 1 | |
** | ** | multiple character operator: ** | infix | 780 | 1 | 1 | |
<> | <> | multiple character operator: <> | infix | 780 | 1 | 1 | |
^ | ^ | circumflex accent | infix | 780 | 1 | 1 | |
′ | ′ | prime | postfix | 800 | 0 | 0 | |
♭ | ♭ | music flat sign | postfix | 800 | 0 | 2 | |
♮ | ♮ | music natural sign | postfix | 800 | 0 | 2 | |
♯ | ♯ | music sharp sign | postfix | 800 | 0 | 2 | |
! | ! | exclamation mark | postfix | 810 | 1 | 0 | |
!! | !! | multiple character operator: !! | postfix | 810 | 1 | 0 | |
// | // | multiple character operator: // | infix | 820 | 1 | 1 | |
@ | @ | commercial at | infix | 825 | 1 | 1 | |
? | ? | question mark | infix | 835 | 1 | 1 | |
ⅅ | ⅅ | double-struck italic capital d | prefix | 845 | 2 | 1 | |
ⅆ | ⅆ | double-struck italic small d | prefix | 845 | 2 | 0 | |
√ | √ | square root | prefix | 845 | 1 | 1 | stretchy |
∛ | ∛ | cube root | prefix | 845 | 1 | 1 | |
∜ | ∜ | fourth root | prefix | 845 | 1 | 1 | |
⁡ | | function application | infix | 850 | 0 | 0 | |
" | " | quotation mark | postfix | 880 | 0 | 0 | accent |
& | & | ampersand | postfix | 880 | 0 | 0 | |
' | ' | apostrophe | postfix | 880 | 0 | 0 | accent |
++ | ++ | multiple character operator: ++ | postfix | 880 | 0 | 0 | |
-- | -- | multiple character operator: -- | postfix | 880 | 0 | 0 | |
^ | ^ | circumflex accent | postfix | 880 | 0 | 0 | stretchy, accent |
_ | _ | low line | postfix | 880 | 0 | 0 | stretchy, accent |
` | ` | grave accent | postfix | 880 | 0 | 0 | accent |
~ | ~ | tilde | postfix | 880 | 0 | 0 | stretchy, accent |
¨ | ¨ | diaeresis | postfix | 880 | 0 | 0 | accent |
ª | ª | feminine ordinal indicator | postfix | 880 | 0 | 0 | accent |
¯ | ¯ | macron | postfix | 880 | 0 | 0 | stretchy, accent |
° | ° | degree sign | postfix | 880 | 0 | 0 | |
² | ² | superscript two | postfix | 880 | 0 | 0 | accent |
³ | ³ | superscript three | postfix | 880 | 0 | 0 | accent |
´ | ´ | acute accent | postfix | 880 | 0 | 0 | accent |
¸ | ¸ | cedilla | postfix | 880 | 0 | 0 | accent |
¹ | ¹ | superscript one | postfix | 880 | 0 | 0 | accent |
º | º | masculine ordinal indicator | postfix | 880 | 0 | 0 | accent |
ˆ | ˆ | modifier letter circumflex accent | postfix | 880 | 0 | 0 | stretchy, accent |
ˇ | ˇ | caron | postfix | 880 | 0 | 0 | stretchy, accent |
ˉ | ˉ | modifier letter macron | postfix | 880 | 0 | 0 | stretchy, accent |
ˊ | ˊ | modifier letter acute accent | postfix | 880 | 0 | 0 | accent |
ˋ | ˋ | modifier letter grave accent | postfix | 880 | 0 | 0 | accent |
ˍ | ˍ | modifier letter low macron | postfix | 880 | 0 | 0 | stretchy, accent |
˘ | ˘ | breve | postfix | 880 | 0 | 0 | accent |
˙ | ˙ | dot above | postfix | 880 | 0 | 0 | accent |
˚ | ˚ | ring above | postfix | 880 | 0 | 0 | accent |
˜ | ˜ | small tilde | postfix | 880 | 0 | 0 | stretchy, accent |
˝ | ˝ | double acute accent | postfix | 880 | 0 | 0 | accent |
˷ | ˷ | modifier letter low tilde | postfix | 880 | 0 | 0 | stretchy, accent |
̂ | ̂ | combining circumflex accent | postfix | 880 | 0 | 0 | stretchy, accent |
̑ | ̑ | combining inverted breve | postfix | 880 | 0 | 0 | accent |
‚ | ‚ | single low-9 quotation mark | postfix | 880 | 0 | 0 | accent |
‛ | ‛ | single high-reversed-9 quotation mark | postfix | 880 | 0 | 0 | accent |
„ | „ | double low-9 quotation mark | postfix | 880 | 0 | 0 | accent |
‟ | ‟ | double high-reversed-9 quotation mark | postfix | 880 | 0 | 0 | accent |
″ | ″ | double prime | postfix | 880 | 0 | 0 | accent |
‴ | ‴ | triple prime | postfix | 880 | 0 | 0 | accent |
‵ | ‵ | reversed prime | postfix | 880 | 0 | 0 | accent |
‶ | ‶ | reversed double prime | postfix | 880 | 0 | 0 | accent |
‷ | ‷ | reversed triple prime | postfix | 880 | 0 | 0 | accent |
‾ | ‾ | overline | postfix | 880 | 0 | 0 | stretchy, accent |
⁗ | ⁗ | quadruple prime | postfix | 880 | 0 | 0 | accent |
⁤ | | invisible plus | infix | 880 | 0 | 0 | |
⃛ | ⃛ | combining three dots above | postfix | 880 | 0 | 0 | accent |
⃜ | ⃜ | combining four dots above | postfix | 880 | 0 | 0 | accent |
⎴ | ⎴ | top square bracket | postfix | 880 | 0 | 0 | stretchy, accent |
⎵ | ⎵ | bottom square bracket | postfix | 880 | 0 | 0 | stretchy, accent |
⏜ | ⏜ | top parenthesis | postfix | 880 | 0 | 0 | stretchy, accent |
⏝ | ⏝ | bottom parenthesis | postfix | 880 | 0 | 0 | stretchy, accent |
⏞ | ⏞ | top curly bracket | postfix | 880 | 0 | 0 | stretchy, accent |
⏟ | ⏟ | bottom curly bracket | postfix | 880 | 0 | 0 | stretchy, accent |
⏠ | ⏠ | top tortoise shell bracket | postfix | 880 | 0 | 0 | stretchy, accent |
⏡ | ⏡ | bottom tortoise shell bracket | postfix | 880 | 0 | 0 | stretchy, accent |
_ | _ | low line | infix | 900 | 1 | 1 |
Several of the following definitions of terms have been borrowed or modified from similar definitions in documents originating from W3C or standards organizations. See the individual definitions for more information.
malignmark
).
sep
and presentation expressions as content. Represents either an identifier
(ci
) or a number (cn
).
mrow
mrow
element that is
"inferred" around the contents of certain layout schemata when
they have other than exactly one argument. Defined precisely in Section 3.1.9 Summary of Presentation Elements<mo>
.
&name;
representing some other data, typically a
non-ASCII character, a sequence of characters, or an external source of
data, e.g. a file containing a set of standard entity definitions such as
ISO Latin 1.
<maligngroup/>
, <malignmark/>
, or the
empty elements none
and
mprescripts
defined in Section 3.4.7 Prescripts and Tensor Indices
<mmultiscripts>
,
<mprescripts/>
,
<none/>
).
The layout schemata are never empty elements (though their content may
contain nothing in some cases), are always expressions, and all allow any
MathML expressions as arguments (except for requirements on argument count,
and the requirement for a certain empty element in mmultiscripts
).
none
or
mprescripts
, or an instance of malignmark
within a token element (defined below);
or a single instance of certain of the content elements
(see Chapter 4 Content Markup for a precise definition of which ones).
apply
element.
mo
element
mo
is defined in
Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
).
malignmark
element.
maction
element)
maction
element (a
layout schema defined in Section 3.7 Enlivening Expressions) that is (at any
given time) "selected" within the viewing state of a MathML
renderer, or by the selection
attribute when the
element exists only in MathML data. Defined precisely in the aforementioned
section.
mrow
elements. Defined precisely in Section 3.2.7 Space <mspace/>
.
<mo>
and in
Section 3.7 Enlivening Expressions.
math
(defined in Section 2.2 The Top-Level
<math>
Element).
The present W3C Math Working Group (2012–2013) is co-chaired by David Carlisle of NAG and Patrick Ion of the AMS; Patrick Ion and and Robert Miner of Design Science were co-chairs 2006-2011. Contact the co-chairs about membership in the Working Group. For the current membership see the W3C Math home page.
Robert Miner, whose leadership and contributions were essential to the development of the Math Working Group and MathML from their beginnings, died tragically young in December 2011.
Participants in the Working Group responsible for MathML 3.0 have been:
All the above persons have been members of the currently chartered Math Working Group, but some not for the whole life of the Working Group. The 22 authors listed for MathML3 at the start of this specification are those who contributed reworkings and reformulations used in the actual text of the specification. Thus the list includes the principal authors of MathML2 much of whose text was repurposed here. They were, of course, supported and encouraged by the activity and discussions of the whole Math Working Group, and by helpful commentary from outside it, both within the W3C and further afield.
For 2003 to 2006 W3C Math Activity comprised a Math Interest Group, chaired by David Carlisle of NAG and Robert Miner of Design Science.
The W3C Math Working Group (2001–2003) was co-chaired by Patrick Ion of the AMS, and Angel Diaz of IBM from June 2001 to May 2002; afterwards Patrick Ion continued as chair until the end of the WG's extended charter.
Participants in the Working Group responsible for MathML 2.0, second edition were:
Earlier active participants of the W3C Math Working Group (2001 – 2003) have included:
The W3C Math Working Group was co-chaired by Patrick Ion of the AMS, and Angel Diaz of IBM from July 1998 to December 2000.
Participants in the Working Group responsible for MathML 2.0 were:
Earlier active participants of this second W3C Math Working Group have included:
At the time of release of MathML 1.0 [MathML1] the Math Working Group was co-chaired by Patrick Ion and Robert Miner, then of the Geometry Center. Since that time several changes in membership have taken place. In the course of the update to MathML 1.01, in addition to people listed in the original membership below, corrections were offered by David Carlisle, Don Gignac, Kostya Serebriany, Ben Hinkle, Sebastian Rahtz, Sam Dooley and others.
Participants in the Math Working Group responsible for the finished MathML 1.0 specification were:
Others who had been members of the W3C Math WG for periods at earlier stages were:
The Working Group benefited from the help of many other people in developing the specification for MathML 1.0. We would like to particularly name Barbara Beeton, Chris Hamlin, John Jenkins, Ira Polans, Arthur Smith, Robby Villegas and Joe Yurvati for help and information in assembling the character tables in Chapter 7 Characters, Entities and Fonts, as well as Peter Flynn, Russell S.S. O'Connor, Andreas Strotmann, and other contributors to the www-math mailing list for their careful proofreading and constructive criticisms.
As the Math Working Group went on to MathML 2.0, it again was helped by many from the W3C family of Working Groups with whom we necessarily had a great deal of interaction. Outside the W3C, a particularly active relevant front was the interface with the Unicode Technical Committee (UTC) and the NTSC WG2 dealing with ISO 10646. There the STIX project put together a proposal for the addition of characters for mathematical notation to Unicode, and this work was again spearheaded by Barbara Beeton of the AMS. The whole problem ended split into three proposals, two of which were advanced by Murray Sargent of Microsoft, a Math WG member and member of the UTC. But the mathematical community should be grateful for essential help and guidance over a couple of years of refinement of the proposals to help mathematics provided by Kenneth Whistler of Sybase, and a UTC and WG2 member, and by Asmus Freytag, also involved in the UTC and WG2 deliberations, and always a stalwart and knowledgeable supporter of the needs of scientific notation.
Changes to the Frontmatter
Changes to Editors and Authors to acknowlege the death of Robert Miner.
Changes to Abstract to highlight MathML may be used in HTML as well as XML.
Update the Status.
Changes to the Chapter 1 Introduction
Changes to Section 1.3 Overview to highlight MathML may be used in HTML as well as XML.
Changes to the Chapter 2 MathML Fundamentals
Changes to Section 2.1.1 General Considerations to highlight MathML may be used in HTML as well as XML.
Add element markup to heading in Section 2.2 The Top-Level
<math>
Element.
Changes to Section 2.1.2 MathML and Namespaces the xmlns
syntax for namespaces only applies to the XML serialisation.
Changes to Section 2.1.5.2 Length Valued Attributes to clarify that values specified with a %
or no unit are multiples of a reference value, which may differ from the default value used when the value is not specified.
Changes to namedspace in Section 2.1.5.2 Length Valued Attributes some attribute values such as "thinmathspace" were marked up as attribute names. (This affected formatting and also the index Section I.2 MathML Attributes).
Additional paragraph describing global document property defaults in Section 2.1.5.4 Default values of attributes
Changes to the Chapter 3 Presentation Markup
Refer to "characters" rather than "MathML Characters" in Section 3.1 Introduction.
Delete the note about bidi in HTML in Section 3.1.5.2 Bidirectional Layout in Token Elements (As there are proposals to change the HTML behavior).
Corrected mistaken refererence to mtext
, replaced by reference to mo
in Section 3.1.8.2 Warning: spacing should not be used to convey meaning
Refer to HTML rather than XHTML in Section 3.1.10 Mathematics style attributes common to presentation elements
Modify heading of Section 3.2.1
Token Element Content Characters, <mglyph/>
.
Note in Section 3.2.1.2 Using images to represent
symbols <mglyph/>
that the requirement to use src
and alt
is not enforced by the schema.
New section Section 3.2.2.2 Embedding HTML in MathML detailing the use of HTML elements on MathML token elements
Use U+2026 rather than . . .
in the example in Section 3.2.3 Identifier <mi>
.
Use percentage lengths rather than unitless lengths in examples in Section 3.2.5.8 Stretching of operators, fences and accents
and Section 3.3.2 Fractions <mfrac>
Reference the Arabic Mathematical Symbols block in describing mathvariant
in Section 3.2.2 Mathematics style attributes common to token elements.
Do not specify that Math defaults may be set by using attributes in the MathML Namespace on the containing document, leave
the mechanism open. Section 3.2.5 Operator, Fence, Separator or Accent
<mo>
Changes to Section 3.2.5.2 Attributes to clarify defaults may be specified in any containing document.
Changes to Section 3.2.5.2.1 Dictionary-based attributes to clarify the interpretation of maxsize
, minsize
and symmetric
values.
Changes to Section 3.2.5.2.3 Indentation attributes to clarify behaviour if indenttarget
results in an unachievable alignment specification.
Changes to the examples in Section 3.2.5.5 Invisible operators so that each example is rendered as a separate math
expression.
Suggest CSS Counters as a possible mechanism for equation numbering in Section 3.5.3 Labeled Row in Table or Matrix
<mlabeledtr>
Minor improvements to the markup in Section 3.3.4 Style Change <mstyle>
.
Minor improvements to the markup in Section 3.3.9 Enclose Expression Inside Notation
<menclose>
.
Changes to the attribute table Section 3.3.6.2 Attributes To clarify that unitless lengths are allowed on mpadded
, meaning, as usual, a multiplier of the stated default. Note that this change also affects the mpadded-length
grammar uin the extracted schema.
Explictly list mprescripts and none in heading for Section 3.4.7 Prescripts and Tensor Indices
<mmultiscripts>
,
<mprescripts/>
,
<none/>
.
Changes to Section 3.5.1.2 Attributes to clarify that displaystyle
defaults to "false" and to clarify the wording describing percentage values for width
.
Minor improvements to the markup in Section 3.6.1 Stacks of Characters <mstack>
.
Editorial improvements to Section 3.6.8.1 Addition and Subtraction.
Modify the markup in the examples in Section 3.6.8.4 Repeating decimal so that MathML renderings are shown in some versions of this specification.
Note that attributes in other namespaces are not available in HTML in Section 3.7.1 Bind Action to Sub-Expression
<maction>
Changes to the Chapter 4 Content Markup
Add element markup to heading in Section 4.2.1.1 Rendering <cn>
,<sep/>
-Represented Numbers .
Add syntax table for qualifier elements in Section 4.3.3.1 Uses of
<domainofapplication>
,
<interval>
,
<condition>
,
<lowlimit>
and
<uplimit>
and Section 4.3.3.2 Uses of <degree>
.
Modify the text in Section 4.1.5 Content MathML Concepts to clarify the role of the Qualifier row of syntax tables. (AM)
Spurious apply
removed from the "0" case in the example in Section 4.4.1.9 Piecewise declaration <piecewise>
, <piece>
, <otherwise>
.
Changes to Rewrite: partialdiffdegree The expression expression-in-x1-xk was rewritten to A. (AM)
Additional note added to the mathmltypes description clarifying that "complex" should be taken as an alias for "complex-cartesian" when rewriting to Strict Content MathML. (AM)
Changes to s_data1.mean, s_dist1.mean, s_dist1.moment and s_data1.moment examples to use new values for ⟨ and ⟩ so the result is in Unicode NFC form.
Changes to markup of syntax tables in Section 4.2.5 Function Application <apply>
and Section 4.2.7.1 The share
element to avoid redundant colspans, which make the html5 version invalid.
Clarify the behavior of qualifiers in Step 4b of the rewrite to Strict Content MathML. (AM)
Clarify that the types of the arguments are used to distinguish between set and multiset use of the set
constructor in Section 4.3.4.1.2 Rewriting to Strict Content MathML and Section 4.3.4.2.2 Rewriting to Strict Content MathML. (AM)
Fix spelling in Section 4.4.2.16 Not <not/>
.
Fix spelling in Section 4.4.3.1 Equals <eq/>
.
Split Section 4.4.7.1 Common trigonometric functions
<sin/>
,
<cos/>
,
<tan/>
,
<sec/>
,
<csc/>
,
<cot/>
into separate sections Section 4.4.7.1 Common trigonometric functions
<sin/>
,
<cos/>
,
<tan/>
,
<sec/>
,
<csc/>
,
<cot/>
,
Section 4.4.7.1 Common trigonometric functions
<sin/>
,
<cos/>
,
<tan/>
,
<sec/>
,
<csc/>
,
<cot/>
,Section 4.4.7.2 Common inverses of trigonometric functions
<arcsin/>
,
<arccos/>
,
<arctan/>
,
<arcsec/>
,
<arccsc/>
,
<arccot/>
, Section 4.4.7.3 Common hyperbolic functions
<sinh/>
,
<cosh/>
,
<tanh/>
,
<sech/>
,
<csch/>
,
<coth/>
, Section 4.4.7.4 Common inverses of hyperbolic functions
<arcsinh/>
,
<arccosh/>
,
<arctanh/>
,
<arcsech/>
,
<arccsch/>
,
<arccoth/>
, add new presentation images for
arcsin.
Add element markup to heading in Section 4.4.7.7 Logarithm <log/>
, <logbase>
.
Minor rearrangement of heading in Section 4.4.8.6 Moment <moment/>
, <momentabout>
Add syntax table for deprecated elements in Section 4.5.1 Declare <declare>
, Section 4.5.3 Relation <fn>
and Section 4.5.2 Relation <reln>
.
Changes to Chapter 5 Mixing Markup Languages for Mathematical Expressions.
Changes to Section 5.1.1 Annotation elements to highlight MathML may be used in HTML as well as XML.
Add additional note warning namespace extensibility example not applicable to HTML.
Add additional note warning namespace extensibility example not applicable to HTML.
Add additional note warning namespace extensibility example not applicable to HTML.
Additional section Section 5.2.3.3 Using annotation-xml
in HTML documents detailing the use of annotation-xml
in HTML documents
Show tag markup around element names in section headings in semantics, annotation and annotation-xml.
Changes to Chapter 6 Interactions with the Host Environment.
Editorial wording changes in Section 6.4 Combining MathML and Other Formats.
Editorial wording changes in Section 6.5 Using CSS with MathML.
Changes to wording on namespace use in Section 6.1 Introduction.
Additional section Section 6.2.2 Recognizing MathML in HTML.
Remove XML Declaration and mml
namespace prefix from the examples in Section 6.3.4 Examples.
Delete recommendation to use prefixed element names in XHTML in Section 6.4.1 Mixing MathML and XHTML.
Split HTML into a separate section from other non-XML use Section 6.4.3 Mixing MathML and HTML and Section 6.4.2 Mixing MathML and non-XML contexts
Remove the reference, Layout engines that lack native MathML support, to [MathMLforCSS] in Chapter 6 Interactions with the Host Environment.
Changes to Chapter 7 Characters, Entities and Fonts.
Change the DTD description in Section 7.3 Entity Declarations to reference the Combined HTML MathML entity set rather than the legacy ISO entity sets. This does not change any existing
definition, but adds the following 38 entity definitions:
"
(U+0022),
&
(U+0026),
<
(U+003C),
>
(U+003E),
©
(U+00A9),
®
(U+00AE),
Α
(U+0391),
Β
(U+0392),
Ε
(U+0395),
Ζ
(U+0396),
Η
(U+0397),
Ι
(U+0399),
Κ
(U+039A),
Μ
(U+039C),
Ν
(U+039D),
Ο
(U+039F),
Ρ
(U+03A1),
Τ
(U+03A4),
Χ
(U+03A7),
ε
(U+03B5),
ο
(U+03BF),
ς
(U+03C2),
ϑ
(U+03D1),
ϒ
(U+03D2),
‌
(U+200C),
‍
(U+200D),
‎
(U+200E),
‏
(U+200F),
‚
(U+201A),
„
(U+201E),
‹
(U+2039),
›
(U+203A),
‾
(U+203E),
⁄
(U+2044),
€
(U+20AC),
™
(U+2122),
ℵ
(U+2135),
↵
(U+21B5).
Changes to Appendix A Parsing MathML.
Modify the schema regular expression to allow the deprecated unitless length attributes.
The schema now enforces a mandatory space and optional minus sign before rownumber in the align
attribute of mtable
and mstack
.
Modify the schema (including DTD and XSD versions) to include the attributes listed in Section 3.2.5.2.3 Indentation attributes on mspace
to match the text description in Section 3.2.7 Space <mspace/>
.
Modify the schema to include the Common Attributes such as id
on bvar
and cn
to match the attribute tables in the text
Modify the regular expressions used for mpadded-length
and length
so that there must be at most one .
and at least one digit. (FW)
New sections: Section A.5 Parsing MathML in XHTML and Section A.6 Parsing MathML in HTML.
Changes to Appendix C Operator Dictionary.
Add entries for the characters listed in Section 7.7.2 Pseudo-scripts.
Changes to Appendix E Working Group Membership and Acknowledgments.
Changes to Section E.1 The Math Working Group Membership to note the death of Robert Miner.
Changes to Appendix G Normative References.
Make [HTML5] normative.
Update the references to current versions, [Bidi], [Entities], [ISO10646], [Normal], [RELAX-NG], [Unicode].
Changes to Appendix H References.
Update the references to current versions, [AAP-math], [Abramowitz1977], [CSS21], [ISO-12083], [XLink].
Changes to Appendix I Index.
Changes to Section I.2 MathML Attributes.
Changes to Chapter 2 MathML Fundamentals.
The attribute href
added to the common MathML attributes, Section 2.1.6 Attributes Shared by all MathML Elements to allow hypertext links.
Additional attributes added to the math
element, see
Section 2.2.1 Attributes.
Changes to Chapter 3 Presentation Markup.
Introduced mechanisms for controlling the Directionality of layout, as described in Section 3.1.5 Directionality.
Introduced mechanisms for controlling linebreaking Section 3.1.7 Linebreaking of Expressions.
Extended mglyph
to support general image inclusion, Section 3.2.1.2 Using images to represent
symbols <mglyph/>
.
The facilities for adjusting spacing with mpadded
have been extended and rationalised, Section 3.3.6 Adjust Space Around Content
<mpadded>
.
Introduced new presentation elements for elementary math layouts, Section 3.6 Elementary Math: mstack
, mlongdiv
, msgroup
, msrow
, mscarries
, mscarry
, and msline
.
Changes to Chapter 4 Content Markup.
Introduced new content elements bind
, share
, cerror
, cs
and cbytes
.
Removed deprecated content elements reln
and fn
.
Removed content element declare
.
The concept of Strict Content MathML and the use of OpenMath Content Dictionaries has been introduced, and the whole chapter restructured.
Changes to Chapter 5 Mixing Markup Languages for Mathematical Expressions.
New Chapter: Chapter 6 Interactions with the Host Environment.
Changes to Chapter 7 Characters, Entities and Fonts.
This chapter is much reduced from the corresponding chapter in previous releases of MathML. All the tables and much of the other content of this chapter is now maintained as a separate document [Entities]
Changes to Appendix A Parsing MathML.
The Normative version of the grammar is now expressed in Relax NG, with DTD and XSD versions being derived.
Three MathML 1 attrbutes on math
that were deprecated and undocumented in MathML2 but retained in the MathML2 DTD have been removed. name
(use id
instead), baseline
and type
(These are not used by any known implementation, so can be removed.) See Chapter 7 of [MathML1].
New Appendix: Appendix B Media Types Registrations.
Changes to Appendix C Operator Dictionary.
The Operator Dictionary table has been updated and rationalised and presented in a new format.
MathML DOM
The chapter and appendices relating to the MathML DOM have been removed from this specification, with the intention of updating them and publishing them as a separate document at a later time.
References to sections in which an element is defined are marked in bold.
In addition to the standard MathML attributes, some attributes from other namespaces such as Xlink or XML Schema are also listed here.