C MathML3 Content Dictionaries

C.1 About Content Dictionaries

The primary role of MathML content elements is to provide a mechanism for recording that a particular notational structure has a particular mathematical meaning. To this end, every content element must have a mathematical definition associated with it in some form. These definitions are provided in the form of content dictionaries, XML files of a certain structure (see . The purpose of this appendix is to provide a default set of content dictionaries for the K-14 fragment of mathematics. Authors may adopt these and come up with additional ones, fixing additional content elements that are encoded via the csymbol element.

Content Dictionaries (CDs) are central to the MathML philosophy of transmitting mathematical information. It is the Content Dictionaries which actually hold the meanings of the objects being transmitted.

For example if application A is talking to application B, and sends, say, an equation involving multiplication of matrices, then A and B must agree on what a matrix is, and on what matrix multiplication is, and even on what constitutes an equation. All this information is held within some Content Dictionaries which both applications agree upon.

Intuitively, a Content Dictionary holds the meanings of (various) mathematical "words", that are represented by symbols in Content MathML

With a set of symbol definitions (perhaps from several Content Dictionaries), A and B can now talk in a common "language".

The concept of a content dictionary has initially been introduced by the OpenMath1 format [OpenMath2000], and has been stabilized and generalized to abstract content dictionaries in the OpenMath2 standard [OpenMath2004], keeping a variant of OpenMath1 CDs as a reference encoding. MathML 3 introduces a content dictionary format that is designed to meet the requirements of the MathML language, while meeting the requirements of OpenMath abstract CDs. We will introduce the format in the next section and then include the MathML3 content dictionaries for the K-14 fragments as part of the MathML3 recommendataion. These CDs are called the MathML3 Recommendation Content Dictionaries

When working with the content elements, it can be useful to keep in mind the following.

• The role of the content elements is analogous to data entry in a mathematical system. The information that is provided is there to facilitate the successful parsing of an expression as the intended mathematical object by a receiving application.

• MathML content elements do not by themselves "perform" any mathematical evaluations or operations. They do not "evaluate" in a browser and any "action" that is ultimately taken on those objects is determined entirely by the receiving mathematical application. For example, editing programs and applications geared to computation for the lower grades would typically leave 3 + 4 as is, whereas computational systems targeting a more advanced audience might evaluate this as 7. Similarly, some computational systems might evaluate sin(0) to 0, whereas others would leave it unevaluated. Yet other computational systems might be unable to deal with pure symbolic expressions like sin(x) and may even regard them as data entry errors. None of this has any bearing on the correctness of the original MathML representation. Where evaluation is mentioned at all in the descriptions below, it is merely to help clarify the meaning of the underlying operation.

• Apart from the instances where there is an explicit interaction with presentation tagging, there is no required rendering (visual or aural) – only a suggested default. As such, the presentations that are included in this section are merely to help communicate to the reader the intended mathematical meaning by association with the same expression written in a more traditional notation.

C.2 The MathML3 Content Dictionary Format

We will now detail the MathML3 Content Dictionary format, on an abstract level and discuss the special case of the MathML Recommendation CDs. Note that the latter are not the only possible ones, any individual or group can set up and publish CDs for the purposes of communication.

Editorial note: MiKo
the material here is copied frm OM-style abstract content dictionaries.

3.3 Content Dictionary: Basic Content Elements

The basic content Elements

C.1 MMLdefinition: interval

Description

This is the MathML constructor element for building an interval on the real line. While an interval can be expressed by combining relations appropriately, they occur here explicitly because of the frequency of their use.

Discussion

The interval element is used to represent simple mathematical intervals of the real number line. It takes an attribute closure, which can take on any of the values "open", "closed", "open-closed", or "closed-open", with a default value of "closed".

A single interval element occuring as the second child of an apply element and preceded by one of the pre-defined n-ary operators is interpreted as a shorthand notation for a domainofapplication. All other uses of an interval element as a child of an apply should be interpreted as ordinary function arguments unless otherwise dictated by the function definition.

More general domains should be constructed using a domainofapplication element or one of the other shortcut notations described in Section 4.2.7 Qualifiers.

The interval element expects two child elements that evaluate to real numbers. or one or more bvar elements and a child element that is a condition defining the interval.

The interval element denotes an interval on the real line with the values represented by its children as end points. The closure attribute is used to qualify the type of interval being represented. For example,

<interval closure="open-closed"><ci>a</ci><ci>b</ci></interval>

represents the open-closed interval often written (a,b].

Classification

constructor

MMLattribute

Name	Value	Default
type	closed | open | open-closed | closed-open	closed

Signature

[type=interval-type](algebraic,algebraic) -> interval(interval-type)

Example

<interval closure="open">
  <ci>x</ci>
  <cn>1</cn>
</interval>

Example

<interval closure="open-closed">
  <cn>0</cn>
  <cn>1</cn>
</interval>

Example

<interval>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><lt/><cn>0</cn><ci>x</ci></apply>
    </condition>
</interval>

Example

<apply><int/>
  <interval><cn>0</cn><cn>1</cn></interval>
  <ci type="function">f</ci>
</apply>

C.2 MMLdefinition: inverse

Description

This MathML element is applied to a function in order to construct a new function that is to be interpreted as the inverse function of the original function. For a particular function F, inverse(F) composed with F behaves like the identity map on the domain of F and F composed with inverse(F) should be an identity function on a suitably restricted subset of the Range of F. The MathML definitionURL attribute should be used to resolve notational ambiguities, or to restrict the inverse to a particular domain or to make it one-sided.

Discussion

The inverse element is applied to a function in order to construct a generic expression nfor the functional inverse of that function. As with other MathML functions, inverse may either be applied to arguments, or it may appear alone, in which case it represents an abstract inversion operator acting on other functions.

A typical use of the inverse element is in an HTML document discussing a number of alternative definitions for a particular function so that there is a need to write and define f^(-1)(x). To associate a particular definition with f^(-1), use the definitionURL and encoding attributes.

Given functions, it is natural to have functional inverses. This is handled by the inverse element.

Functional inverses can be problematic from a mathematical point of view in that they implicitly involve the definition of an inverse for an arbitrary function F. Even at the K-through-12 level the concept of an inverse F ^-1 of many common functions F is not used in a uniform way. For example, the definitions used for the inverse trigonometric functions may differ slightly depending on the choice of domain and/or branch cuts.

MathML adopts the view: if F is a function from a domain D to D', then the inverse G of F is a function over D' such that G(F(x)) = x for x in D. This definition does not assert that such an inverse exists for all or indeed any x in D, or that it is single-valued anywhere. Also, depending on the functions involved, additional properties such as F(G(y)) = y for y in D' may hold.

The inverse element is applied to a function whenever an inverse is required. For example, application of the inverse sine function to x, i.e. sin^-1 (x), is encoded as:

    <apply> <apply> <inverse/> <sin/> </apply> <ci> x </ci> </apply> 

While arcsin is one of the predefined MathML functions, an explicit reference to sin^-1(x) might occur in a document discussing possible definitions of arcsin.

Classification

operator

Signature

(function) -> function

[definitionURL=URI](function) -> function(definition)

Property

ForAll( y, such y in domain( f^(-1) ), f( f^(-1)(y) ) = y

          
<apply><forall/>
  <bvar><ci>y</ci></bvar>
  <bvar><ci type="function">f</ci></bvar>
  <condition>
    <apply><in/>
      <ci>y</ci>
      <apply><csymbol definitionURL="domain"><mtext>Domain</mtext></csymbol>
        <apply><inverse/><ci type="function">f</ci></apply>
      </apply>
    </apply>
  </condition>
  <apply><eq/>
    <apply><ci type="function">f</ci>
      <apply><apply><inverse/><ci type="function">f</ci></apply>
        <ci>y</ci>
      </apply>
    </apply>
    <ci>y</ci>
  </apply>
</apply>

Example

<apply><inverse/>
  <sin/>
</apply>

Example

<apply><inverse definitionURL="www.example.com/MathML/Content/arcsin"/>
  <sin/>
</apply>

Example

<apply>
  <apply><inverse/>
    <ci type="matrix"> a </ci>
  </apply>
  <ci> A </ci>
</apply>

C.3 MMLdefinition: lambda

Description

This is the operation of lambda calculus that constructs a function from an expression and a variable. Lambda is an n-ary function, where all but an optional domain of application and the last argument are bound variables and the last argument is an expression possibly involving those variables. The lambda function can be viewed as the inverse of function application.

For example, Lambda( x, F ) is written as \lambda x [F] in the lambda calculus literature. The expression F may contain x but the full lambda expression is regarded to be free of x. A computational application receiving a MathML lambda expression should not evaluate x or test for x. Such an application may apply the lambda expression as a function to arguments in which case any result that is computed is computed through parameter substitutions into F.

Note that a lambda expression on an arbitrary function applied to a the bound variable is equivalent to that arbitrary function. A domain of application can be used to restrict the defined function to a specific domain.

Discussion

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 lambda element is used to construct a user-defined function from an expression. The last child is an expression defining the function in terms of the bound variables declared by the bvar and any domainofapplication (see Section 4.2.7 Qualifiers) elements coming before it. The last element is typically an apply element, but can also be any container element. The following constructs (x, sin x)

<lambda>
  <bvar><ci>x</ci></bvar>
  <apply><sin/><ci>x</ci></apply>
</lambda>

The following constructs the constant function (x, 3)

<lambda><bvar><ci> x </ci></bvar><cn>3</cn></lambda>

The lambda calculus allows a user to construct a function from a variable and an expression. For example, the lambda construct underlies the common mathematical idiom illustrated here:

Let f be the function taking x to x² + 2

There are various notations for this concept in mathematical literature, such as (x, F(x)) = F or (x, [F]) =F, where x is a free variable in F.

This concept is implemented in MathML with the lambda element. A lambda construct with n (possibly 0) internal variables is encoded by a lambda element, where the first n children are bvar elements containing the identifiers of the internal variables. This is followed by an optional domainofapplication qualifier (see Section 4.2.7 Qualifiers) and an expression defining the function. The defining expression is typically an apply, but can also be any expression.

The following constructs (x, sin(x+1)):

<lambda>
  <bvar><ci> x </ci></bvar>
  <apply>
    <sin/>
    <apply>
      <plus/>
      <ci> x </ci>
      <cn> 1 </cn>
    </apply>
  </apply>
</lambda>

To use declare and lambda to construct the function f for which f( x) = x² + x + 3 use:

<declare type="function">
  <ci> f </ci>
  <lambda>
    <bvar><ci> x </ci></bvar>
    <apply>
      <plus/>
      <apply>
        <power/>
        <ci> x </ci>
        <cn> 2 </cn>
      </apply>
      <ci> x </ci>
      <cn> 3 </cn>
    </apply>
  </lambda>
</declare>

The following markup declares and constructs the function J such that J(x, y) is the integral from x to y of t⁴ with respect to t.

<declare type="function">
  <ci> J </ci>
  <lambda>
    <bvar><ci> x </ci></bvar>
    <bvar><ci> y </ci></bvar>
    <apply> <int/>
      <bvar><ci> t </ci></bvar>
      <lowlimit><ci> x </ci></lowlimit>
      <uplimit><ci> y </ci></uplimit>
      <apply>
        <power/>
        <ci>t</ci>
        <cn>4</cn>
      </apply>
    </apply>
  </lambda>
</declare>

The function J can then in turn be applied to an argument pair.

Classification

constructor

Signature

(bvar*,anything) -> function

(domainofapp,function) -> function

(bvar+,domainofapp,anything) -> function

Property

ForAll( F, lambda(x,F(x)) = F )

          
<apply><forall/>
  <bvar><ci>F</ci></bvar>
  <apply><eq/>
    <lambda>
      <bvar><ci>x</ci></bvar>
      <apply><ci>F</ci><ci>x</ci></apply>
    </lambda>
    <ci>F</ci>
  </apply>
</apply>

Example

a simple lambda construct.

<lambda>
  <bvar><ci> x </ci></bvar>
  <apply><sin/>
    <apply>
      <plus/>
      <ci> x </ci>
      <cn> 1 </cn>
    </apply>
  </apply>
</lambda>

Example

The next example constructs a one-argument function in which the argument b specifies the upper bound of a specific definite integral.

<lambda>
  <bvar><ci> b </ci></bvar>
  <apply>
    <int/>
    <bvar><ci> x </ci></bvar>
    <lowlimit><ci> a </ci></lowlimit>
    <uplimit><ci> b </ci></uplimit>
    <apply>
       <fn><ci> f </ci></fn>
       <ci> x </ci>
    </apply>
  </apply>
</lambda>

Example

The domainofapplication child restricts the possible values of the arguments of the constructed function. For instance, the following two lambda constructs are representations of a function on the integers.

<lambda>
  <bvar><ci> x </ci></bvar>
  <domainofapplication><integers/></domainofapplication>
  <apply><sin/><ci> x </ci></apply>
</lambda>

Example

If a lambda construct does not contain bound variables, then the arity of the constructed function is unchanged, and the lambda construct is redundant, unless it also contains a domainofapplication construct that restricts existing functional arguments, as in this example, which is a variant representation for the function above. In particular, if the last child of a lambda construct is not a function, say a number, then the lambda construct will not be a function, but the same number. Of course, in this case a domainofapplication does not make sense

<lambda>
  <domainofapplication><integers/></domainofapplication>
  <sin/>
</lambda>

C.4 MMLdefinition: compose

Description

This is the MathML constructor for composing functions. In order for a composition to be meaningful, the range of the first function should be the domain of the second function, etc. . However, since no evaluation takes place in MathML, such a construct can safely be used to make statements such as that f composed with g is undefined.

The result is a new function whose domain is the domain of the first function and whose range is the range of the last function and whose definition is equivalent to applying each function to the previous outcome in turn as in:

(f @ g )( x ) == f( g(x) ).

This function is often denoted by a small circle infix operator.

Discussion

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 an n-ary operator. As an n-ary operator, therefore it may take qualifiers.

Classification

function

Signature

(function*) -> function

Property

ForAll( x, (f@g)(x) = f(g(x) )

          
<apply><forall/>
  <bvar><ci>x</ci></bvar><bvar><ci>f</ci></bvar><bvar><ci>g</ci></bvar>   
  <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>
</apply>

Example

The use of fn is deprecated. Use type="function" instead.

          
<apply><compose/>
  <ci type="function"> f </ci>
  <ci type="function"> g </ci>
  <sin/>
</apply>

Example

<apply>
  <compose/>
  <ci type="function"> f </ci>
  <ci type="function"> g </ci>
  <ci type="function"> h </ci>
</apply>

Example

<apply>
  <apply><compose/>
    <fn><ci> f </ci></fn>
    <fn><ci> g </ci></fn>
  </apply>
  <ci> x </ci>
</apply>

Example

<apply>
  <fn><ci> f </ci></fn>
  <apply>
    <fn><ci> g </ci></fn>
    <ci> x </ci>
  </apply>
</apply>

C.5 MMLdefinition: ident

Description

The ident element represents the identity function. MathML makes no assumption about the function space in which the identity function resides. Proper interpretation of the domain (and hence codomain) of the identity function depends on the context in which it is used.

Classification

constructor

Signature

function

Property

ForAll( x, ident(x) = x )

          
	  
<apply><forall/>
  <bvar><ci>x</ci></bvar>
  <apply><eq/>
    <apply><ident/><ci>x</ci></apply>
    <ci>x</ci>
  </apply>
</apply>

Example

<apply>
  <eq/>
  <apply><compose/>
    <fn><ci> f </ci></fn>
    <apply><inverse/>
      <fn><ci> f </ci></fn>
    </apply>
  </apply>
  <ident/>
</apply>

Example

<apply><eq/>
  <apply><compose/>
    <ci type="function"> f </ci>
    <apply><inverse/><ci type="function"> f </ci></apply>
  </apply>
  <ident/>
</apply>

C.6 MMLdefinition: domain

Description

The domain element denotes the domain of a given function, which is the set of values over which it is defined.

Discussion

The domain element denotes the domain of a given function, which is the set of values over which it is defined.

Classification

function

Signature

(function) -> set

Example

If f is a function from the reals to the rationals, then:

<apply>
  <eq/>
  <apply><domain/>
    <fn><ci> f </ci></fn>
  </apply>
  <reals/>
</apply>

C.7 MMLdefinition: codomain

Description

The codomain (range) element denotes the codomain of a given function, which is a set containing all values taken by the function. The codomain may contain additional points which are not realized by applying the the function to elements of the domain.

Discussion

The codomain element denotes the codomain of a given function, which is a set containing all values taken by the function. It is not necessarily the case that every point in the codomain is generated by the function applied to some point of the domain. (For example I may know that a function is integer-valued, so its codomain is the integers, without knowing (or stating) which subset of the integers is mapped to by the function.)

Codomain is sometimes also called Range.

Classification

function

Signature

(function) -> set

Property

ForAll( y, Exists(x,y =f(x)), member(y,codomain(f)) )

Example

If f is a function from the reals to the rationals, then:

	  

<apply>
  <eq/>
  <apply><codomain/>
    <fn><ci> f </ci></fn>
  </apply>
  <rationals/>
</apply>

C.8 MMLdefinition: image

Description

The image element denotes the image of a given function, which 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.

Discussion

The image element denotes the image of a given function, which 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.

Classification

function

Signature

(function) -> set

Property

ForAll( x, x in image(f), ThereExists(y,f(y)=x) )

Example

The real sin function is a function from the reals to the reals, taking values between -1 and 1.

	  
<apply><eq/>
  <apply><image/><sin/></apply>
  <interval><cn>-1</cn><cn> 1</cn></interval>
</apply>

Example

<apply><forall/>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><in/>
      <ci>x</ci>
      <apply><image/><ci>f</ci></apply>
    </apply>
  </condition>
  <apply><in/>
    <ci>x</ci>
    <apply><codomain/><ci>f</ci></apply>
  </apply>
</apply>

C.9 MMLdefinition: piecewise

Description

The piecewise, piece, and otherwise elements are used to support 'piecewise' declarations of the form H(x) = 0 if x less than 0, H(x) = 1 otherwise. The piece and otherwise elements describe evaluation rules. If no rule applies or if more than one rule applies but they give different answers then the expression is undefined.

Discussion

The piecewise, piece, and otherwise elements are used to support "piecewise" declarations 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.

otherwise 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.

The piecewise elements are constructors.

The piecewise, piece, otherwise elements are used to support "piecewise" declarations of the form " H(x) = 0 if x less than 0, H(x) = x otherwise".

<piecewise>
  <piece>
      <cn>0</cn>
      <apply><lt/><ci>x</ci><cn>0</cn></apply>
  </piece>
  <otherwise><ci>x</ci></otherwise>
</piecewise>

Classification

constructor

Signature

(piece*, otherwise?) -> algebraic

(piece*, otherwise?) -> anything

Property

ForAll( x, x in domain(f), the evaluation rules collectively produce at most one value in codomain(f) )

Example

<piecewise>
  <piece><cn> 0</cn><apply><lt/><ci> x</ci> <cn> 0</cn></apply></piece>
  <otherwise><ci>x</ci></otherwise>
</piecewise>

Example

The value of the abs function evaluated at x can be written as:

          
<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>

C.10 MMLdefinition: piece

Description

The piece element is used to construct the conditionally defined values as part of a piecewise object.

Classification

constructor

Signature

(algebraic, boolean) -> piece

(anything, boolean) -> piece

Example

<piecewise>
  <piece><cn>0</cn><apply><lt/><ci> x</ci> <cn> 0</cn></apply></piece>
  <otherwise><ci>x</ci></otherwise>
</piecewise>

C.11 MMLdefinition: otherwise

Description

The otherwise element is used to describe the value of a piecewise construct when none of the conditions of the associated pieces are satisfied.

Classification

constructor

Signature

(algebraic) -> otherwise

(anything) -> otherwise

Example

<piecewise>
  <piece><cn> 0</cn><apply><lt/><ci> x</ci> <cn> 0</cn></apply></piece>
  <otherwise><ci>x</ci></otherwise>
</piecewise>

3.4 Content Dictionary: Arithmetic Algebra and Logic

C.1 MMLdefinition: quotient

Description

quotient is the binary function used to represent the operation of integer division. quotient(a,b) denotes q such that a = b*q+r, with |r| less than |b| and a*r non-negative.

Discussion

The quotient element is the operator used for division modulo a particular base. When the quotient operator is applied to integer arguments a and b, the result is the "quotient of a divided by b". That is, quotient returns the unique integer q such that a = q b + r. (In common usage, q is called the quotient and r is the remainder.)

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	integer

Signature

(integer, integer) -> integer

Property

ForAll( [a,b], b != 0, a = b*quotient(a,b) + rem(a,b) )

      <apply><forall/>
  <bvar><ci>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <condition><apply><neq/><ci>b</ci><cn>0</cn></apply></condition>
  <apply><eq/>
    <ci>a</ci>
    <apply><plus/>
      <apply><times/>
          <ci>b</ci>
          <apply><quotient/><ci>a</ci><ci>b</ci></apply>
      </apply>
      <apply><rem/><ci>a</ci><ci>b</ci></apply>
    </apply>
  </apply>
</apply>

Example

Various mathematical applications will use this data in different ways. Editing applications might choose an image such as shown below, while a computationally based application would evaluate it to 2 whena=13 and b=5.

	  
	  <apply><quotient/> <ci> a </ci> <ci> b </ci> </apply> 
	

Example

<apply>
  <quotient/>
  <cn>5</cn>
  <cn>4</cn>
</apply>

C.2 MMLdefinition: factorial

Description

This is the unary operator used to construct factorials. Factorials are defined by n! = n*(n-1)* ... * 1

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	integer

Signature

(algebraic) -> algebraic

(integer) -> integer

Property

ForAll( n, n \gt 0, n! = n*(n-1)! )

          
<apply><forall/>
  <bvar><ci>n</ci></bvar>
  <condition><apply><gt/><ci>n</ci><cn>0</cn></apply></condition>
  <apply><eq/>
    <apply><factorial/><ci>n</ci></apply>
    <apply><times/>
      <ci>n</ci>
      <apply><factorial/>
        <apply><minus/><ci>n</ci><cn>1</cn></apply>
      </apply>
    </apply>
  </apply>
</apply>

Property

0! = 1

          
<apply><eq/>
  <apply><factorial/><cn>0</cn></apply>
  <cn>1</cn>
</apply>

Example

If this were evaluated at n = 5 it would evaluate to 120.

		 
		<apply><factorial/>
  <ci>n</ci>
</apply>

C.3 MMLdefinition: divide

Description

This is the binary MathML operator that is used indicate the mathematical operation a "divided by" b.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(algebraic, algebraic) -> algebraic

(complex, complex) -> complex

(real, real) -> real

(rational, rational) -> rational

(integer, integer) -> rational

Property

Division by Zero error

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <apply><eq/>
    <apply><divide/><ci> a </ci><cn> 0 </cn></apply>
    <notanumber/>
  </apply>
</apply>

Property

ForAll( a, a!= 0, a/a = 1 )

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <condition><apply><neq/><ci>a</ci><cn>0</cn></apply></condition>
  <apply><eq/>
    <apply><divide/><ci>a</ci><ci>a</ci></apply>
    <cn>1</cn>
  </apply>
</apply>

Example

As a MathML expression, this does not evaluate. However, on receiving such an expression, some applications may attempt to evaluate and simplify the value. For example, when a=5 and b=2 some mathematical applications may evaluate this to 2.5 while others will treat is as a rational number.

	    
<apply><divide/>
  <ci> a </ci>
  <ci> b </ci>
</apply>

C.4 MMLdefinition: max

Description

This is the n-ary operator used to represent the maximum of a set of elements. The elements may be listed explicitly or they may be described by a domainofapplication, for example, the maximum over all x in the set A. The domainofapplication is often abbreviated by placing a condition directly on a bound variable.

The max operator is a n-ary operators may use the domain of application qualifiers as described in Section 4.4.10 Lifted Associative Commutative Operators. For example, the min and max functions accept a bvar schema in cases where the maximum or minimum is being taken over a set of values specified by a condition schema together with an expression to be evaluated on that set. In MathML1.0, the bvar element was optional when using a condition; if a condition element containing a single variable was given by itself following a min or max operator, the variable was implicitly assumed to be bound, and the expression to be maximized or minimized (if absent) was assumed to be the single bound variable. This usage is deprecated in MathML 2.0 in favor of explicitly stating the bound variable(s) and the expression to be maximized or minimized in all cases.

The min and max elements may also be applied to a list of values in which case no qualifier schemata are used.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(algebraic*) -> algebraic

(domainofapp,function) -> algebraic

(bvar+,domainofapp,algebraic) -> algebraic

Property

ForAll( x in S, max(y in S,y) \geq x )

Example

Maximum of a finite listing of elements

          
<apply><max/><cn>2</cn><cn>3</cn><cn>5</cn></apply>

Example

Max(y^3, y in (0,1))

          
<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>

		

Example

When the objects are to be compared explicitly they are listed as arguments to the function as in:

		
		<apply>
		  <max/>
		  <ci> a </ci>
		  <ci> b </ci>
		</apply>
		

Example

In MathML1.0, the bound variable and expression to be evaluated (x) could be omitted in the example below: this usage is deprecated in MathML2.0 in favor of explicitly stating the bound variable and expression in all cases:

		  
			
			<apply>
			  <max/>
			  <bvar><ci>x</ci></bvar>
			  <condition>
			    <apply><and/>
			      <apply><in/><ci>x</ci><ci type="set">B</ci></apply>
			      <apply><notin/><ci>x</ci><ci type="set">C</ci></apply>
			    </apply>
			  </condition>
			  <ci>x</ci>
			</apply>
			

C.5 MMLdefinition: min

Description

This is the n-ary operator used to represent the minimum of a set of elements. The elements may be listed explicitly or they may be described by a condition, e.g., the minimum over all x in the set A.

The elements must all be comparable if the result is to be well defined.

The min operator is a are n-ary operators may use the domain of application qualifiers as described in Section 4.4.10 Lifted Associative Commutative Operators. For example, the min and max functions accept a bvar schema in cases where the maximum or minimum is being taken over a set of values specified by a condition schema together with an expression to be evaluated on that set. In MathML1.0, the bvar element was optional when using a condition; if a condition element containing a single variable was given by itself following a min or max operator, the variable was implicitly assumed to be bound, and the expression to be maximized or minimized (if absent) was assumed to be the single bound variable. This usage is deprecated in MathML 2.0 in favor of explicitly stating the bound variable(s) and the expression to be maximized or minimized in all cases.

The min and max elements may also be applied to a list of values in which case no qualifier schemata are used.

Discussion

The element min is used to compare the values of its arguments. It returns the minimum of these values respectively.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(algebraic*) -> algebraic

(domainofapp,function) -> algebraic

(bvar+,domainofapp,anything) -> algebraic

Example

Minimum of a finite listing of elements

          
<apply><min/><cn>2</cn><cn>3</cn><cn>5</cn></apply>

Example

min(y^2, y in (0,1))

          
<apply>
  <min/>
  <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>2</cn></apply>
</apply>

Example

The elements to be compared may also be described using bound variables with a condition element and an expression to be minimized, as in:

Note that the bound variable must be stated even if it might be implicit in conventional notation.

		
		<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>
		

C.6 MMLdefinition: minus

Description

This is the subtraction operator for an additive group.

If one argument is provided this operator constructs the additive inverse of that group element. If two arguments, say a and b, are provided it constructs the mathematical expression a - b.

Discussion

The minus element is the subtraction operator.

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).

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(real) -> real

(algebraic) -> algebraic

(real,real) -> real

(algebraic,algebraic) -> algebraic

[type=MathMLtype](MathMLtype) -> MathMLtype

[type=MathMLtype](MathMLtype,MathMLtype) -> MathMLtype

(set,set) -> set

(multiset,multiset) -> multiset

Property

ForAll( x, x-x=0 )

          
<apply><forall/>
  <bvar><ci> x </ci></bvar>
  <apply><eq/>
    <apply><minus/><ci> x </ci><ci> x </ci></apply>
    <cn>0</cn>
  </apply>
</apply>

Example

If this were evaluated at x=5 and y=2 it would yield 3.

		
		
		<apply> <minus/>
		  <ci> x </ci>
		  <ci> y </ci>
		</apply>
		

Example

<apply><minus/>
  <cn>3</cn>
</apply>

C.7 MMLdefinition: plus

Description

This is the n-ary addition operator of an algebraic structure. Ordinarily, the operands are provided explicitly. As an n-ary operation the operands can also be generated by allowing a function or expression vary over a domain of application though the sum element is normally used for that purpose. If no operands are provided, the expression represents the additive identity. If one operand, a, is provided the expression evaluates to "a". If two or more operands are provided, the expression represents the (semi) group element corresponding to a left associative binary pairing of the operands. The meaning of mixed operand types not covered by the signatures shown here are left up to the target system.

To use different type coercion rules different from those indicated by the signatures, use the definitionURL attribute to identify a new definition.

Discussion

The plus element is the addition operator.

As an n-ary operator, the operands may in principle also be provided using a domain of application as described in Section 4.4.10 Lifted Associative Commutative Operators. However, such expressions can already be represented explicitly using sum so the plus does not normally take qualifiers.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

[type=MathMLtype](anything*) -> MathMLtype

(set*) -> set

(multiset*) -> multiset

(algebraic*) -> algebraic

(real*) -> real

(complex*) -> complex

(integer*) -> integer

(domainofapp,function) -> algebraic

(bvar+,domainofapp,algebraic) -> algebraic

Property

an sum of no terms is 0

          
<apply><eq/>
  <apply><plus/></apply>
  <cn>0</cn>
</apply> 

Property

a sum of one term is equal to itself

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <apply><eq/>
    <apply><plus/><ci>a</ci></apply>
    <cn>a</cn>
  </apply> 
</apply>

Property

Commutativity

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <condition>
    <apply><and/>
      <apply><in/><ci>a</ci><reals/></apply>
      <apply><in/><ci>b</ci><reals/></apply>
    </apply>
  </condition>
  <apply><eq/>
    <apply><plus/><ci>a</ci><ci>b</ci></apply>
    <apply><plus/><ci>b</ci><ci>a</ci></apply>
  </apply>
</apply>

Example

<apply><plus/>
  <cn>3</cn>
</apply>

Example

<apply><plus/>
  <cn>3</cn>
  <cn>5</cn>
</apply>

Example

If this were evaluated at x = 5, y = 2 and z = 1 it would yield 8.

	   
	   
			<apply>
			  <plus/>
			  <ci> x </ci>
			  <ci> y </ci>
			  <ci> z </ci>
			</apply>
			

C.8 MMLdefinition: power

Description

This is the binary powering operator that is used to construct expressions such as a "to the power of" b. In particular, it is the operation for which a "to the power of" 2 is equivalent to a * a.

Discussion

The power element is a generic exponentiation operator. That is, when applied to arguments a and b, it returns the value of "a to the power of b".

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(algebraic, algebraic) -> algebraic

(complex, complex) -> complex

(real, real) -> complex

(rational, integer) -> rational

(integer, integer) -> rational

[type=MathMLtype](anything,anything) -> MathMLtype

Property

ForAll( a, a!=0, a^0=1 )

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <condition><apply><neq/><ci>a</ci><cn>0</cn></apply></condition>
  <apply><eq/>
    <apply><power/><ci>a</ci><cn>0</cn></apply>
    <cn>1</cn>
  </apply>
</apply>

Property

ForAll( a, a^1=a )

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <apply><eq/>
    <apply><power/><ci>a</ci><cn>1</cn></apply>
    <ci>a</ci>
  </apply>
</apply>

Property

ForAll( a, 1^a=1 )

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <apply><eq/>
    <apply><power/><cn>1</cn><ci>a</ci></apply>
    <cn>1</cn>
  </apply>
</apply>

Example

<apply><power/><cn>2</cn><ci>x</ci></apply>

Example

<apply><power/><ci> x </ci><cn> 3 </cn></apply>

Example

If this were evaluated at x= 5 it would yield 125.

	  
	  
		<apply>
		  <power/>
		  <ci> x </ci>
		  <cn> 3 </cn>
		</apply>
		

C.9 MMLdefinition: rem

Description

This is the binary operator used to represent the integer remainder a mod b. For arguments a and b, such that a = b*q + r with |r| < |b| it represents the value r.

Discussion

The rem element is the operator that returns the "remainder" of a division modulo a particular base. When the rem operator is applied to integer arguments a and b, the result is the "remainder of a divided by b". That is, rem returns the unique integer, r such that a = q b+ r, where r < q. (In common usage, q is called the quotient and r is the remainder.)

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	integer

Signature

(integer, integer) -> integer

[type=MathMLtype](MathMLtype,MathMLtype) -> MathMLtype

Property

rem(a, 0) is undefined

Property

ForAll( [a,b], b!=0, a = b*quotient(a,b) + rem(a,b))

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <condition><apply><neq/><ci>b</ci><cn>0</cn></apply></condition>
  <apply><eq/>
    <ci>a</ci>
    <apply><plus/>
      <apply><times/>
        <ci>b</ci>
  <apply><quotient/><ci>a</ci><ci>b</ci></apply>
      </apply>
    <apply><rem/>
    <ci>a</ci>
    <ci>b</ci>
  </apply>
</apply>
</apply>
</apply>

Example

If this were evaluated at a = 15 and b = 8 it would yield 7.

		
		<apply><rem/><ci> a </ci><ci> b </ci></apply>

C.10 MMLdefinition: times

Description

This is the n-ary multiplication operator of a ring. Ordinarily, the operands are provided explicitly. As an n-ary operation the operands can also be generated by allowing a function or expression vary over a domain of application though the product element is normally used for that purpose. If no arguments are supplied then this represents the multiplicative identity. If one argument is supplied, this represents an expression that would evaluate to that single argument.

Discussion

The times element is the n-ary multiplication operator.The operands are usually listed explicitly. As an n-ary operator, the operands may in principle also be provided using a domain of application as described in Section 4.4.10 Lifted Associative Commutative Operators. However, such expressions can already be represented explicitly by using product so the times does not normally take qualifiers.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(algebraic*) -> algebraic

(complex*) -> complex

(real*) -> real

(rational*) -> rational

(integer*) -> integer

(domainofapp,function) -> algebraic

(bvar+,domainofapp,anything) -> algebraic

Property

ForAll( [a,b], condition(in({a,b}, Commutative)), a*b=b*a )

Property

ForAll( [a,b,c], Associative, a*(b*c)=(a*b)*c ), associativity

Property

multiplicative identity

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <apply><eq/>
    <apply><times/><cn>1</cn><ci>a</ci></apply>
    <ci>a</ci>
  </apply> 
</apply>

Property

a*0=0

Property

Commutative property

          
<apply><forall/>
  <bvar><ci>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <condition>
    <apply><and/>
      <apply><in/><ci>a</ci><reals/></apply>
      <apply><in/><ci>b</ci><reals/></apply>
    </apply>
  </condition>
  <apply><eq/>
    <apply><times/><ci>a</ci><ci>b</ci></apply>
    <apply><times/><ci>b</ci><ci>a</ci></apply>
  </apply>
</apply>

Property

a*0=0

	  
	    <apply><forall/>
  <bvar><ci>a</ci></bvar>
  <apply><eq/>
    <apply><times/><cn>0</cn><ci>a</ci></apply>
    <cn>0</cn>
  </apply> 
</apply>
	

Example

If this were evaluated at a = 5.5 and b = 3 it would yield 16.5.

	  
	  <apply><times/><ci>a</ci><ci>b</ci></apply>
	

C.11 MMLdefinition: root

Description

This is the binary operator used to construct the nth root of an expression. The first argument "a" is the expression and the second object "n" denotes the root, as in ( a ) ^ (1/n)

Discussion

The root element is used to construct roots. The kind of root to be taken is specified by a "degree" child, 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.

Note that MathML3 supports a degree element in the legacy container representation (see Section 4.4.8 degree for details). If no degree is present, a default value of 2 is used.

Classification

function

MMLattribute

Name	Value	Default
type	real | complex | principal_branch | MathMLType	real

Signature

(algebraic) -> root(degree(2),algebraic)

(anything) -> root(degree(2),anything)

(degree,anything) -> root

Property

ForAll( bvars(a,n), root(degree(n),a) = a^(1/n) )

Example

The nth root of a is is given by

          
<apply><root/>
  <degree><ci> n </ci></degree>
  <ci> a </ci>
</apply>

C.12 MMLdefinition: gcd

Description

This is the n-ary operator used to construct an expression which represents the greatest common divisor of its arguments. If no argument is provided, the gcd is 0. If one argument is provided, the gcd is that argument.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	integer

Signature

[type=MathMLtype](MathMLtype*) ->MathMLtype

(integer*) -> integer

(domainofapp,function) -> algebraic

(bvar+,domainofapp,algebraic) -> algebraic

Property

<apply><forall/>
<forall/>  
  <bvar><ci>x</ci></bvar>
  <apply><eq/>
    <apply><gcd/>
      <ci>x</ci>
      <cn>1</cn>
    </apply>
    <cn>1</cn>
  </apply>
</apply>

Example

<apply><gcd/>
  <cn>12</cn>
  <cn>17</cn>
</apply>

Example

<apply><gcd/>
  <cn>3</cn>
  <cn>5</cn>
  <cn>7</cn>
</apply>

Example

If this were evaluated at a = 15, b = 21, c = 48, it would yield 3

		
		
		<apply> <gcd/>
		  <ci> a </ci>
		  <ci> b </ci>
		  <ci> c </ci>
		</apply>
		

C.13 MMLdefinition: and

Description

This is the n-ary logical "and" operator. It is used to construct the logical expression which were it to be evaluated would have a value of "true" when all of its operands have a truth value of "true", and "false" otherwise.

Discussion

The and element is the Boolean "and" operator.

As an n-ary operator, its operands may also be generated as described inSection 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	boolean

Signature

(boolean*) -> boolean

Property

ForAll( p, (true and p = p) )

Property

ForAll( [p,q], (p and q = q and p) )

Property

x and not(x) = false

Example

If this were evaluated and both a and b had truth values of "true", then the result would be "true".

		
		<apply><and/>
		  <ci>a</ci>
		  <ci>b</ci>
		  </apply>
		

C.14 MMLdefinition: or

Description

The is the n-ary logical "or" operator. The constructed expression has a truth value of true if at least one of its arguments is true.

Discussion

The or element is the Boolean "or" operator.

As an n-ary operator, its operands may also be generated as described inSection 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	boolean

Signature

(boolean*) -> boolean

[type="boolean"](symbolic*) -> boolean

Example

<apply>
  <or/>
  <ci> a </ci>
  <ci> b </ci>
</apply>

C.15 MMLdefinition: xor

Description

The is the n-ary logical "xor" operator. The constructed expression has a truth value of true if an odd number of its arguments are true.

Discussion

As an n-ary operator, its operands may also be generated as described inSection 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	boolean

Signature

(boolean*) -> boolean

[type="boolean"](symbolic*) -> symbolic

Property

x xor x = false

Property

x xor not(x) = true

Example

<apply>
  <xor/>
  <ci> a </ci>
  <ci> b </ci>
</apply>

C.16 MMLdefinition: not

Description

This is the unary logical "not" operator. It negates the truth value of its single argument. e.g., not P is true when P is false and false when P is true.

Discussion

The not operator is the Boolean"not" operator.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	boolean

Signature

(boolean) -> boolean

[type="boolean"](algebraic) -> boolean

Example

<apply>
  <not/>
  <ci> a </ci>
</apply>

C.17 MMLdefinition: implies

Description

This is the binary "implies" operator. It is used to construct the logical expression "A implies B".

Discussion

The implies element is the Boolean relational operator "implies".

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	boolean

Signature

(boolean,boolean) -> boolean

Property

false implies x

Example

Mathematical applications designed for the evaluation of such expressions would evaluate this to "true" when a = "false" and b = "true".

		
		<apply>
  <implies/>
  <ci> A </ci>
  <ci> B </ci>
</apply>

C.18 MMLdefinition: forall

Description

The forall operator is the logical "For all" quantifier. The bound variables, if any, appear first and are tagged using the bvar element. Next comes an optional condition on the bound variables. The last argument is the boolean expression that is asserted to be true for all values of the bound variables that meet the specified conditions (if any).

Discussion

The forall element represents the universal quantifier of logic. It is usually used in conjunction with one or more bound variables, an optional condition element, and an assertion. It may also be used with a domain of application and function as described in Section 4.2.7 Qualifiers in which case the assertion corresponds to applying the function to an element of the specified domain., which should take the form of an apply element.

The universal quantifier forall is used in conjunction with one or more bvar schemata to represent simple logical assertions. There are two main main ways of using the logical quantifier operators. The first usage is for representing a simple, quantified assertion. For example, the statement "there exists x < 9" would be represented as:

<bind>
  <exists/>
  <bvar><ci> x </ci></bvar>
  <apply><lt/><ci>x</ci><cn>9</cn></apply>
</bind>

The second usage is for representing implications. Hypotheses are given by a condition element following the bound variables. For example the statement "for all x < 9, x < 10" would be represented as:

<bind>
  <forall/>
  <bvar><ci>x</ci></bvar>
  <condition><apply><lt/><ci>x</ci><cn>9</cn></apply></condition>
  <apply><lt/><ci>x</ci><cn>10</cn></apply>
</bind>

Note that in both these usages one or more bvar qualifiers are mandatory.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	boolean

Signature

(domainofapp,function) -> boolean

(bvar+,domainofapp?,boolean) -> boolean

Example

The example encodes a simple identity.

		
		<apply>
		  <forall/>
		  <bvar><ci> x </ci></bvar>
		  <apply><eq/>
		    <apply>
		      <minus/><ci> x </ci><ci> x </ci>
		    </apply>
		    <cn>0</cn>
		  </apply>
		</apply>

Example

The example is more involved, and makes use of an optional condition element.

		
		
		<apply>
		  <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>
		</apply>
		

Example

The example uses both the forall and exists quantifiers.

		
		
		<apply>
		  <forall/>
		  <bvar><ci> n </ci></bvar>
		  <condition>
		    <apply><and/>
		      <apply><gt/><ci> n </ci><cn> 0 </cn></apply>
		      <apply><in/><ci> n </ci><integers/></apply>
		    </apply>
		  </condition>
		  <apply>
		    <exists/>
		    <bvar><ci> x </ci></bvar>
		    <bvar><ci> y </ci></bvar>
		    <bvar><ci> z </ci></bvar>
		    <condition>
		      <apply><and/>
		        <apply><in/><ci> x </ci><integers/></apply>
		        <apply><in/><ci> y </ci><integers/></apply>
		        <apply><in/><ci> z </ci><integers/></apply>
		      </apply>
		    </condition>
		    <apply>
		      <eq/>
		      <apply>
		        <plus/>
		        <apply><power/><ci> x </ci><ci> n </ci></apply>
		        <apply><power/><ci> y </ci><ci> n </ci></apply>
		      </apply>
		      <apply><power/><ci> z </ci><ci> n </ci></apply>
		    </apply>
		  </apply>
		</apply>
		

Example

<apply>
		  <forall/>
		  <bvar><ci> x </ci></bvar>
		  <condition><apply><lt/><ci> x </ci><cn> 0
		  </cn></apply></condition>
		  <ci> x </ci>
</apply>

C.19 MMLdefinition: exists

Description

This is the MathML operator that is used to assert existence, as in "There exists an x such that x is real and x is positive."

- The first argument indicates the bound variable,

- The second optional argument places conditions on that bound variable.

- The last argument is the expression that is asserted to be true.

Discussion

The exists element represents the existential quantifier of logic. Typically, it is used in conjunction with one or more bound variables, an optional condition element, and an assertion, which may take the form of either an apply or reln element.The exists element may also be used with a general domain of application and function as described in Section 4.2.7 Qualifiers. For such uses the assertion is obtained by applying the function to an element of the specified domain.

The universal quantifier forall is used in conjunction with one or more bvar schemata to represent simple logical assertions. There are two main main ways of using the logical quantifier operators. The first usage is for representing a simple, quantified assertion. For example, the statement "there exists x < 9" would be represented as:

<bind>
  <exists/>
  <bvar><ci> x </ci></bvar>
  <apply><lt/><ci>x</ci><cn>9</cn></apply>
</bind>

The second usage is for representing implications. Hypotheses are given by a condition element following the bound variables. For example the statement "for all x < 9, x < 10" would be represented as:

<bind>
  <forall/>
  <bvar><ci>x</ci></bvar>
  <condition><apply><lt/><ci>x</ci><cn>9</cn></apply></condition>
  <apply><lt/><ci>x</ci><cn>10</cn></apply>
</bind>

Note that in both these usages one or more bvar qualifiers are mandatory.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	boolean

Signature

(bvar+,boolean) -> boolean

(bvar+,domainofapp,anything) -> boolean

Example

The following example encodes the sense of the expression "there exists an x such that f(x) = 0".

		
		
		<apply><exists/>
		  <bvar><ci>x</ci></bvar>
		  <apply><eq/>
		    <apply><ci>f</ci>
		      <ci>x</ci>
		    </apply>
		    <cn>0</cn>
		  </apply>
		</apply>

Example

<apply>
  <exists/>
  <bvar><ci> x </ci></bvar>
  <domainofapplication>
    <ci type="set">C</ci>
  </domainofapplication>
  <apply>
    <eq/>
    <apply>
      <power/><ci>x</ci><cn>2</cn>
    </apply>
    <cn>4</cn>
  </apply>
</apply>

C.20 MMLdefinition: abs

Description

A unary operator which represents the absolute value of its argument. In the complex case this is often referred to as the modulus.

Discussion

The abs element represents the absolute value of a real quantity or the modulus of a complex quantity.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(algebraic) -> algebraic

(real) -> real

(complex) -> real

Property

for all x and y, abs(x) + abs(y) >= abs(x+y)

Example

The following example encodes the absolute value of x.

		
		
<apply><abs/><ci>x</ci></apply>

C.21 MMLdefinition: conjugate

Description

The unary "conjugate" arithmetic operator is used to represent the complex conjugate of its argument.

Discussion

The conjugate element represents the complex conjugate of a complex quantity.

Classification

function

Signature

(algebraic) -> algebraic

(complex) -> complex

Example

The following example encodes the conjugate of x + iy.

	  
	  <apply><conjugate/>
 <apply><plus/>
    <ci> x </ci>
    <apply><times/>
      <cn> &ImaginaryI; </cn>
    <ci> y </ci>
    </apply>
  </apply>
</apply>

C.22 MMLdefinition: arg

Description

The unary "arg" operator is used to construct an expression which represents the "argument" of a complex number.

Discussion

The arg operator gives the "argument" of a complex number, which is the angle (in radians) it makes with the positive real axis. Real negative numbers have argument equal to + .

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(complex) -> real

Example

The following example encodes the argument operation onx + iy.

		
		<apply><arg/>
  <apply><plus/>
    <ci> x </ci>
    <apply><times/><imaginaryi/><ci>y</ci></apply>
  </apply>
</apply>

C.23 MMLdefinition: real

Description

A unary operator used to construct an expression representing the "real" part of a complex number.

Discussion

The real operator (introduced in MathML 2.0) gives the real part of a complex number, that is the x component in x + i y

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(complex) -> real

Property

ForAll( [x,y], x in R, Y in R, real(x+i*y)=x) )

          
<apply><forall/>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <condition>
    <apply><and/>
      <apply><in/><ci>x</ci><reals/></apply>
      <apply><in/><ci>y</ci><reals/></apply>
    </apply>
  </condition>
  <apply><eq/>
    <apply><real/>
      <apply><plus/>
        <ci> x </ci>
        <apply><times/><imaginaryi/><ci>y</ci></apply>
      </apply>
    </apply>
    <ci> x </ci>
  </apply>
</apply>

Example

The following example encodes the real operation onx + iy.A MathML-aware evaluation system would return the x component, suitably encoded.

		
		
<apply><real/>
  <apply><plus/>
    <ci> x </ci>
    <apply><times/><imaginaryi/><ci>y</ci></apply>
  </apply>
</apply>

C.24 MMLdefinition: imaginary

Description

The unary function used to construct an expression which represents the imaginary part of a complex number.

Discussion

The imaginary operator (introduced in MathML 2.0) gives the imaginary part of a complex number, that is, the y component in x + i y.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(complex) -> real

Property

ForAll( [x,y], Imaginary(x + i*y) = y )

          
<apply><forall/>
  <bvar><ci type="real"> x </ci></bvar>
  <bvar><ci type="real"> y </ci></bvar>
  <apply><eq/>
    <apply><imaginary/>
      <apply><plus/>
        <ci type="real"> x </ci>
        <apply><times/><imaginaryi/><ci>y</ci></apply>
      </apply>
    </apply>
    <ci type="real"> y </ci>
  </apply>
</apply>

Example

The following example encodes the imaginary operation on x + iy.A MathML-aware evaluation system would return the y component, suitably encoded.

		
		<apply><imaginary/>
  <apply><plus/>
    <ci> x </ci>
    <apply><times/><imaginaryi/><ci>y</ci></apply>
  </apply>
</apply>

C.25 MMLdefinition: lcm

Description

This n-ary operator is 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.

Discussion

The lcm element (introduced in MathML 2.0) is used to denote the lowest common multiple of its arguments.

As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	integer

Signature

[type=MathMLtype](MathMLtype*) -> MathMLtype

(integer*) -> integer

(algebraic*) -> algebraic

(domainofapp,function) -> algebraic

(bvar+,domainofapp,anything) -> algebraic

Property

ForAll( x, lcm(x,1)=x )

          
<apply><forall/>
  <bvar><ci>x</ci></bvar>
  <apply><eq/>
    <apply><lcm/><ci>x</ci><cn>1</cn></apply>
    <ci>x</ci>
  </apply>
</apply>

Example

<apply><lcm/>
  <cn>12</cn>
  <cn>17</cn>
</apply>

Example

<apply><lcm/>
  <cn>3</cn>
  <cn>5</cn>
  <cn>7</cn>
</apply>

Example

If this were evaluated at a = 2, b = 4, c = 6 it would yield 12.

		
		
		<apply> <lcm/>
		  <ci> a </ci>
		  <ci> b </ci>
		  <ci> c </ci>
		</apply>
		

C.26 MMLdefinition: floor

Description

The floor element is used to denote the round-down (towards -infinity) operator.

Discussion

The floor element (introduced in MathML 2.0) is used to denote the round-down (towards -infinity) operator.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	integer

Signature

(real) -> integer

[type=MathMLtype](algebraic) -> algebraic

Property

ForAll( x, floor(x) <= x )

          <apply><forall/>
  <bvar><ci>x</ci></bvar>
  <apply><leq/>
    <apply><floor/>
      <ci>x</ci>
    </apply>
    <ci>x</ci>
  </apply>
</apply>

Example

If this were evaluated at a = 15.015, it would yield 15.

		
		<apply> <floor/>
  <ci> a </ci>
</apply>

Example

		<apply> <forall/>
		  <bvar><ci> a </ci></bvar>
		  <apply><and/>
		    <apply><leq/>
		    <apply><floor/>
		    <ci>a</ci>
		    </apply>
		    <ci>a</ci>
		  </apply>    
		    <apply><lt/>
		      <ci>a</ci>
		    <apply><plus/>
		      <apply><floor/>
		      <ci>a</ci>
		    </apply>
		    <cn>1</cn>
		    </apply>
		  </apply>
		  </apply>
		</apply>
		

C.27 MMLdefinition: ceiling

Description

The ceiling element is used to denote the round-up (towards +infinity) operator.

Discussion

The ceiling element (introduced in MathML 2.0) is used to denote the round-up (towards +infinity) operator.

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	integer

Signature

(real) -> integer

[type=MathMLtype](algebraic) -> algebraic

Property

ForAll( x, ceiling(x) >= x )

          <apply><forall/>
  <bvar><ci>x</ci></bvar>
  <apply><geq/>
    <apply><ceiling/>
      <ci>x</ci>
    </apply>
    <ci>x</ci>
  </apply>
</apply>

Example

If this were evaluated at a = 15.015, it would yield 16.

		
		<apply> <ceiling/>
  <ci> a </ci>
</apply>

Example

		<apply> <forall/>
		  <bvar><ci> a </ci></bvar>
		  <apply><and/>
		    <apply><lt/>
		    <apply><minus/>
		      <apply><ceiling/>
		      <ci>a</ci>
		    </apply>
		    <cn>1</cn>
		    </apply>
		      <ci>a</ci>
		  </apply>
		    <apply><leq/>
		    <ci>a</ci>
		    <apply><ceiling/>
		    <ci>a</ci>
		    </apply>
		  </apply>    
		  </apply>
		</apply>
		

3.5 Content Dictionary: Relations

C.1 MMLdefinition: eq

Description

This n-ary function is used to indicate that two or more quantities are equal. There must be at least two arguments.

Discussion

The eq element is the relational operator "equals".

As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

nary relation

Signature

(real,real+) -> boolean

(boolean, boolean+) -> boolean

(set,set+) -> set

(multiset,multiset+) -> multiset

(domainofapp,function) -> boolean

(bvar+,domainofapp,anything) -> boolean

Property

Symmetric

Property

Transitive

Property

Reflexive

Example

<apply><eq/>
  <cn type="rational">2<sep/>4</cn><cn type="rational">1<sep/>2</cn>
</apply>

Example

<apply><eq/><ci type="set">A</ci><ci type="set">B</ci></apply>

Example

<apply><eq/><ci type="multiset">A</ci><ci type="multiset">B</ci></apply>

Example

If this were tested at a = 5.5 and b = 6 it would yield the truth value false.

      <apply><eq/><ci> a </ci><ci> b </ci></apply>

C.2 MMLdefinition: neq

Description

This binary relation represents the relation "not equal to" which returns true unless the two arguments are equal.

Discussion

The neq element is the "not equal to" relational operator.

The neq element is a binary relation.

Classification

function

Signature

(real,real) -> boolean

(boolean,boolean) -> boolean

(set,set) -> set

(multiset,multiset) -> multiset

Property

Symmetric

Example

<apply><neq/><cn>3</cn><cn>4</cn></apply>

Example

If this were tested at a = 5.5 and b = 6 it would yield the truth value true.

		
		  
			<apply>
			  <neq/>
			  <ci> a </ci>
			  <ci> b </ci>
			</apply>
			

C.3 MMLdefinition: gt

Description

This n-ary function represents the relation "greater than" which returns true if each argument in turn is greater than the one following it. There must be at least two arguments.

Discussion

The gt element is the "greater than" relational operator.

The gt element is an n-ary relation. As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

Signature

(real,real+) -> boolean

(domainofapp,function) -> boolean

(bvar+,domainofapp,algebraic) -> boolean

Property

Transitive

Example

<apply><gt/><cn>3</cn><cn>2</cn></apply>

Example

If this were tested at a = 5.5 and b = 6 it would yield the truth value false.

	  
	<apply><gt/><ci>a</ci><ci>b</ci></apply>

C.4 MMLdefinition: lt(role: application)

Description

This n-ary function represents the relation "less than" which returns true if each argument in turn is less than the one following it. There must be at least two arguments.

Discussion

The lt element is the "less than" relational operator.

The lt element is an n-ary relation. As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

Signature

(real,real+) -> boolean

( domainofapp , function ) -> boolean

( bvar+ , domainofapp , algebraic) -> boolean

Property

Transitive

Example

<apply><lt/><cn>2</cn><cn>3</cn><cn>4</cn></apply>

Example

If this were tested at a = 5.5 and b = 6 it would yield the truth value "true".

		
		
		<apply>
		  <lt/>
		  <ci> a </ci>
		  <ci> b </ci>
		</apply>
		

C.5 MMLdefinition: geq

Description

This element represents the n-ary greater than or equal to function. which returns true if each argument in turn is greater than or equal to the one following it. . There must be at least two arguments.

Discussion

The geq element is the relational operator "greater than or equal".

The geq element is an n-ary relation. As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

Signature

(real,real+) -> boolean

( domainofapp , function ) -> boolean

( bvar+ , domainofapp , algebraic ) -> boolean

Property

Transitive

Property

Reflexive

Example

<apply><geq/><cn>4</cn><cn>3</cn><cn>3</cn></apply>

Example

If this were tested for a = 5.5 and b = 5.5 it would yield the truth value true.

		
		
		<apply>
		  <geq/>
		  <ci> a </ci>
		  <ci> b </ci>
		</apply>
		

C.6 MMLdefinition: leq

Description

This n-ary function represents the relation "less than or equal to" which returns true if each argument in turn is less or equal to the one following it. There must be at least two arguments.

Discussion

The leq element is the relational operator "less than or equal".

The leq element is an n-ary relation. As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

Signature

(real,real+) -> boolean

( domainofapp , function ) -> boolean

( bvar+ , domainofapp , arithmetic ) -> boolean

Property

Transitive

Property

Reflexive

Example

<apply><leq/><cn>3</cn><cn>3</cn><cn>4</cn></apply>

Example

If a = 5.4 and b = 5.5 this will yield the truth value true.

		
		
		<apply>
		  <leq/>
		  <ci> a </ci>
		  <ci> b </ci>
		</apply>
		

C.7 MMLdefinition: equivalent

Description

This element represents the n-ary logical equivalence function in which two boolean expressions are said to be equivalent if their truth values are equal for all choices of values of the boolean variables appearing in them.

Discussion

The equivalent element is the "equivalence" relational operator.

Its operands may be generated by allowing a function or expression to vary over a domain of application. Therefore it may take qualifiers.

Classification

nary relation

Signature

(boolean,boolean+) -> boolean

(domainofapp,function) -> boolean

(bvar+,domainofapp,boolean) -> boolean

Property

Symmetric

Property

Transitive

Property

Reflexive

Example

This yields the truth value true for all values of a.

		
		
		<apply>
		  <equivalent/>
		  <ci> a </ci>
		  <apply>
		    <not/>
		    <apply> <not/> <ci> a </ci> </apply>
		  </apply>
		</apply>
		

C.8 MMLdefinition: approx

Description

This element is used to indicate that two or more quantities are approximately equal. If a more precise definition of approximately equal is required the definintionURL should be used to identify a suitable definition.

Discussion

The approx element is the relational operator "approximately equal". This is a generic relational operator and no specific arithmetic precision is implied

Classification

binary relation

MMLattribute

Name	Value	Default
type	MathMLType	real

Signature

(real,real+) -> boolean

(domainofapp,function) -> boolean

(bvar+,domainofapp,boolean) -> boolean

Property

Symmetric

Property

Transitive

Property

Reflexive

Example

<apply><approx/><pi/><cn type="rational">22<sep/>7</cn></apply>

C.9 MMLdefinition: factorof

Description

This is the binary MathML operator that is used indicate the mathematical relationship a "is a factor of" b. This relationship is true just if b mod a = 0

Discussion

The factorof element is the relational operator element on two integers a and b specifying whether one is an integer factor of the other.

The factorof element is an binary relational operator

Classification

function

MMLattribute

Name	Value	Default
type	MathMLType	integer

Signature

(integer, integer) -> boolean

Property

ForAll( [a,b], a and b integers, a divides b if there is an integer c such that a*c = b )

Example

<apply><factorof/>
  <ci> a </ci>
  <ci> b </ci>
</apply>

3.6 Content Dictionary: Calculus and Vector Calculus

C.1 MMLdefinition: int

Description

The indefinite integral of a function.

Discussion

The int element is the operator element for an integral.

Classification

function

Signature

(function) -> function

Example

The integral over the sine function is cosine function.

      
<apply><eq/><apply><int/><sin/></apply><cos/></apply>
    

C.2 MMLdefinition: int

Description

The definite integral of a function over a domain of application.

Discussion

The defint element is the operator element for function over a domain of integration.

Classification

function

Signature

(set,function) -> function

Example

This example specifies an interval of the real line as the domain of integration with an interval element. In this form the integrand is provided as a function and no mention is made of a bound variable..

      
<apply><defint/>
  <interval><ci>a</ci><ci>b</ci></interval>
  <cos/>
</apply>
    

Example

<apply><defint/><ci type="set">D</ci><ci type="function">f</ci></apply>

C.3 MMLdefinition: int

Description

The definite (without condition or indefinite (with) integral of an algebraic expression with respect to a bound variable.

Discussion

The Int element is the operator element for an integral, bound variables serve as the integration variables and definite integrals are indicated by providing a domain of integration.

Classification

function

Signature

(bvar,algebraic) -> algebraic

(bvar+,domainofapp,algebraic) -> algebraic

Example

This example specifies an interval of the real line as the domain of integration with an interval element. In this form the integrand is provided as a function and no mention is made of a bound variable..

      
<apply><Int/>
 <interval><ci>a</ci><ci>b</ci></interval>
 <cos/>
 </bind>
    

Example

This example specifies the integrand using an expression involving a bound variable and an interval

      
<bind><Int/>
  <bvar><ci>x</ci></bvar>
  <domainofapplication><apply><interval/><ci>a</ci><ci>b</ci></apply></domainofapplication>
  <apply><cos/><ci>x</ci></apply>
</bind>
    

Example

<bind><Int/>
  <bvar><ci> x </ci></bvar>
  <condition><apply><in/><ci>x</ci><ci type="set">D</ci></apply></condition>
  <apply><ci type="function">f</ci><ci>x</ci></apply>
</bind>

C.4 MMLdefinition: diff

Description

The diff element is the differentiation operator element for functions of a single variable.

Discussion

diff is applied directly to an actual function such as sine or cosine, thereby denoting a function which is the derivative of the original function, as in

<apply><diff/><ci type="function>f</ci></apply>

If diff is given a number n as a second argument, then it is interpreted as a n-fold differentiation operator.

<apply><ndiff/><cn>2</cn><ci type="function>f</ci></apply>

Classification

function

Signature

function -> function

Example

diff(sin) = cos

          
	<apply><eq/><apply><diff/><sin/></apply><cos/></apply>

Example

diff (2,sin) = sin

          
	<apply><eq/><apply><ndiff/><cn>2</cn><sin/></apply><sin/></apply>

C.5 MMLdefinition: Diff

Description

Diff is the algebraic differentiation operator.

Discussion

Issue algebraic_diff	wiki (member only)   ISSUE-10 (member only)
Algebraic differentiation operator
In the diff operator, we used a bvar schema to specify the bound variables. But in the new canonical MathML, we only allow bvar in bind (which carries the expectation of alpha-equivalence). Thus we cannot make algebraic diff a binding operator, and have to pass the differentiation variable as a regular argument (i.e. no bvar For seeing how things work out in the concert of things, I have added an Diff symbol to the content dictionary calculus_veccalc for algebraic differentiation. Have a look, how it works.
Resolution	None recorded

Diff is the binding operator for operator takes two (or three) arguments: an expression involving a single variable such as sin(x), or cos(x) or a polynomial in x and the variable modulo which the derivative is being taken. For instance:

<bind><Diff/>
  <bvar><ci>x</ci></bvar>
  <apply><ci>f</ci><ci>x</ci></apply><ci>x</ci>
</bind>

If Diff is given a number n as a third argument, then it is interpreted as a n-fold algebraic differentiation operator.

Issue algebraic_ndiff	wiki (member only)   ISSUE-10 (member only)
nfold algebraic differentiation
I have some trouble representing the iterated differentiation operator. I actually think that the degree should be something that is a parameter to the binding operator, so the following would be the most natural representation. <bind> <apply><Diff/><cn>3</cn></apply> <bvar><ci>x</ci></bvar> <apply><ci>f</ci><ci>x</ci></apply> </apply> But this messes up the cdrole classification in OpenMath. Rather than a binder, the Diff operator is now of role application, but Diff(3) is of role binder. I do not see any way around introducing more complex roles (or relegating the role checking to the type systym e.g. STS.). I am pretty sure that this problem will also appear somewhere else. Maybe we can re-interpret the role binding as "has to be at the head of an application of a bind object. This might actually work, let's keep our fingers crossed that we will not come into even more complex representational situations.
Resolution	None recorded

Classification

binding constructor

Signature

(var,algebraic) -> algebraic

Property

ForAll( [x,n], n!=0, diff( x^n , x ) = n*x^(n-1) )

Example

diff( sin(x) , x ) = cos

          
<apply><eq/>
  <bind><Diff/>
    <bvar><ci>x</ci></bvar>
    <apply><sin/><ci>x</ci></apply>
  </bind>
  <apply><cos/><ci>x</ci></apply>
</apply>

Example

Diff(x^2,x)

          
<apply><Diff/>
  <apply><power/><ci>x</ci><cn>2</cn></apply>
  <ci>x</ci>
</apply>

Example

Diff(f(x),x)

The derivative with respect to x of an expression in x such as f (x) can be written as:

          
<apply><Diff/>
  <apply><ci type="function">f</ci><ci>x</ci></apply>
  <ci>x</ci>
</apply>

Example

anDiff( sin(x) , x, 2) = sin(x)

          
<apply><eq/>
  <apply><Diff/>
    <apply><sin/><ci>x</ci></apply>
    <ci>x</ci>
    <cn>2</cn>
  </apply>
  <apply><sin/><ci>x</ci></apply>
</apply>

C.6 MMLdefinition: partialdiff

Description

This symbol is used to express partial differentiation. It occurs in two forms: one form corresponding to the differentiation of algebraic expressions (often displayed using the Leibnitz notation), and the other to express partial derivatives of actual functions (often expressed as $D_{1,2} f $ )

For the first form, the arguments are the bound variables followed by the algebraic expression. The result is an algebraic expression. Repetitions of the bound variables are indicated using the degree element. The total degree is indicated by use of a degree element at the top level.

For the second form, there are two arguments: a list of indices indicating by position which coordinates are involved in constructing the partial derivatives, and the actual function. The coordinates may be repeated.

Discussion

The partialdiff element is the partial differentiation operator element for functions or algebraic expressions in several variables.

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.

For the expression case the actual variable is designated by a bvar element that is a child of the containing apply element. The bvar elements may also contain a degree element, which specifies the order of the derivative to be taken.

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.

For the case of partial differentiation of a function, the containing apply takes two child elements: firstly a list of indices indicating by position which coordinates are involved in constructing the partial derivatives, and secondly the actual function to be partially differentiated. The coordinates may be repeated.

The partialdiff operator accepts zero or more bvar schemata, and an optional degree qualifier schema. The bvar schema specify, in order, the variables with respect to which the derivative is being taken. Each bvar element may contain a degree schema which is used to specify the order of the derivative being taken with respect to that variable. The optional degree schema qualifier associated with the partialdiff element itself (that is, appearing as a child of the enclosing apply element rather than of one of the bvar qualifiers) is used to represent the total degree of the differentiation. Each degree schema used with partialdiff is expected to contain a single child schema. For example,

<bind><partialdiff/>
  <bvar>
    <degree><cn>2</cn></degree>
    <ci>x</ci>
  </bvar>
  <bvar><ci>y</ci></bvar>
  <bvar><ci>x</ci></bvar>
  <degree><cn>4</cn></degree>
  <ci type="function">f</ci>
</apply>

denotes the mixed partial derivative ( d⁴ / d²x dy dx ) f.

Classification

function

MMLattribute

Name	Value	Default
type	function | algebraic	algebraic

Signature

(bvar+,degree?,algebraic) -> algebraic

(vector,function) -> function

Property

ForAll( [x,y], partialdiff( x * y , x ) = partialdiff(x,x)*y + partialdiff(y,x)*x )

Property

ForAll( [x,a,b], partialdiff( a + b , x ) = partialdiff(a,x) + partialdiff(b,x) )

Example

d^k/(dx^m dy^n) f(x,y)

          
	  <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>

Example

d^2/(dx dy) f(x,y)

          
	  <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>

Example

D_{1,1,3}(f)

          
	  <apply><partialdiff/>
  <list><cn>1</cn><cn>1</cn><cn>3</cn></list>
  <ci type="function">f</ci>
</apply>

C.7 MMLdefinition: divergence

Description

This symbol is used to represent the divergence function.

Given, one argument which is a vector of scalar valued functions defined on the coordinates x_1, x_2, ... x_n. It returns a scalar value function. That function satisfies the defining relation:

divergence(F) = \partial(F_(x_1))/\partial(x_1) + ... + \partial(F_(x_n))/\partial(x_n)

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.

Discussion

The divergence element is the vector calculus divergence operator, often called div.

Classification

unary function

Signature

(vector(function)) -> function

(bvar+,vector(algebraic)) -> algebraic

Example

<apply>
  <divergence/>
  <ci> a </ci>
</apply>

Example

<apply><divergence/><ci type="vector"> E</ci></apply>

Example

	  <declare><ci>F</ci><vector><ci>f1</ci><ci>f2</ci><ci>f3</ci></vector></declare>
<apply><divergence/><ci>F</ci></apply>
	

Example

<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>

Example

If a is a vector field defined inside a closed surface S enclosing a volume V, then the divergence of a is given by

	  
<apply>
  <eq/>
  <apply><divergence/><ci type="vectorfield">a</ci></apply>
  <apply>
    <limit/>
    <bvar><ci> V </ci></bvar>
    <condition>
      <apply>
        <tendsto/>
        <ci> V </ci>
        <cn> 0 </cn>
      </apply>
    </condition>
    <apply>
      <divide/>
      <apply>
        <int encoding="text" definitionURL="SurfaceIntegrals.htm"/>
        <bvar><ci> S</ci></bvar>
        <ci> a </ci>
      </apply>
      <ci> V </ci>
    </apply>
  </apply>
</apply>
	

C.8 MMLdefinition: grad

Description

The gradient element is the vector calculus gradient operator, often called grad. It represents the operation that constructs a vector of partial derivatives vector( df/dx_1 , df/dx_2, ... df/dx_n )

Discussion

The grad element is the vector calculus gradient operator, often called grad.

Classification

unary function

Signature

(function) -> vector(function)

(bvar+,algebraic) -> vector(algebraic)

Example

Where for example f is a scalar function of three real variables.

	   
	  <apply><grad/><ci type="function"> f</ci></apply>	
	

Example

<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>

C.9 MMLdefinition: curl

Description

This symbol is used to represent the curl operator. It requires both a coordinates and a vector of expressions defined over those coordinates. It returns a vector valued expression.

In its functional form the coordinates are implicit in the definition of the function so it needs only one argument which is a vector valued function and returns a vector of functions.

Given F = F(x,y,z) = ( f1(x,y,z) , f2(x,y,z), f3(x,y,z) ) and coordinate names (x,y,z) the following relationship must hold:

curl(F) = i X \partial(F)/\partial(x) + j X \partial(F)/\partial(y) + j X \partial(F)/\partial(Z) where i,j,k are the unit vectors corresponding to the x,y,z axes respectivly and the multiplication X is cross multiplication.

Discussion

The curl element is the vector calculus curl operator.

Classification

unary function

Signature

(bvar,bvar,bvar,algebraic) -> vector(algebraic)

(vector(function) ) -> vector(function)

Property

curl(F) = i X \partial(F)/\partial(x) + j X \partial(F)/\partial(y) + j X \partial(F)/\partial(Z)

Example

Where for example a is a vector field.

	  
	  <apply><curl/><ci>a</ci></apply>
	

Example

<apply><curl/><ci type="vector">f</ci></apply>

C.10 MMLdefinition: laplacian

Description

This is the element used to indicate an application of the laplacian operator. It may be applied directly to expressions, in which case the coordinate names are provided in order by use of bvar. It may also be applied directly to a function F in which case, the definition below is for F = F(x_1, x_2, ... x_n) where x_1, x_2, ... x_n are the coordinate names.

laplacian(F) = \partial^2(F)/\partial(x_1)^2 + ... + \partial^2(F)/\partial(x_n)^2

Discussion

The laplacian element is the vector calculus laplacian operator.

Classification

unary function

Signature

(bvar+,algebraic) -> algebraic

(scalar_valued_function) -> scalar_valued_function

Example

<apply><laplacian/><ci type="vector"> E</ci></apply>

Example

<declare><ci>F</ci><vector><ci>f1</ci><ci>f2</ci><ci>f3</ci></vector></declare>
<apply><laplacian/><ci>F</ci></apply>

Example

<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>

Example

Where for examplef is a scalar function of three real variables.

	<apply>
  <eq/>
  <apply><laplacian/>
    <ci> f </ci>
  </apply>
  <apply>
    <divergence/>
    <apply><grad/>
      <ci> f </ci>
    </apply>
  </apply>
</apply>

3.7 Content Dictionary: Theory of Sets

C.1 MMLdefinition: set

Description

The set element is the container element that constructs a set of elements. They may be explicitly listed, or defined by expressions or functions evaluated over a domain of application. The domain of application may be given explicitly, or provided by means of one of the shortcut notations.

Discussion

Editorial note: MiKo
Take the list and set parts of the discussion apart and make the MMLdefinitions selfcontained, maybe move some of the discussion into the CD description

The set element is the container element that constructs a set of elements. The elements of a set can be defined either by explicitly listing the elements, or by evaluating a function over a domain of application as described in Section 4.2.7 Qualifiers.

The type attribute indicates type of the set. Predefined values: "normal", "multiset". "multiset" indicates that repetitions are allowed. The default value is "normal".

Issue type_on_set	wiki (member only)
Keep type attribute on set?
It seems very awkward to keep this, it would probably be better to have an multiset content dictionary as OpenMath does.
Resolution	None recorded

The child elements of a possibly empty list element are the actual components of an ordered list. For example, an ordered list of the three symbols a, b, and c is encoded as

<list><ci>a</ci><ci>b</ci><ci>c</ci></list>

Sets and lists can also be constructed by evaluating a function over a domain of application, each evaluation corresponding to a term of the set or list. In the most general form a domain is explicitly specified by a domainofapplication element together with optional bvar elements. Qualifications involving a domainofapplication element can be abbreviated in several ways as described in Section 4.2.7 Qualifiers. For example, a bvar and a condition element can be used to define lists where membership depends on satisfying certain conditions.

An order attribute can be used to specify what ordering is to be used. When the nature of the child elements permits, the ordering defaults to a numeric or lexicographic ordering.

Sets are structured much the same as lists except that there is no implied ordering and the type of set may be "normal" or "multiset" with "multiset" indicating that repetitions are allowed.

For both sets and lists, the child elements must be valid MathML content elements. The type of the child elements is not restricted. For example, one might construct a list of equations, or of inequalities.

The child elements of a possibly empty list element are the actual components of an ordered list. For example, an ordered list of the three symbols a, b, and c is encoded as

<list><ci>a</ci><ci>b</ci><ci>c</ci></list>

Sets and lists can also be constructed by evaluating a function over a domain of application, each evaluation corresponding to a term of the set or list. In the most general form a domain is explicitly specified by a domainofapplication element together with optional bvar elements. Qualifications involving a domainofapplication element can be abbreviated in several ways as described in Section 4.2.7 Qualifiers. For example, a bvar and a condition element can be used to define lists where membership depends on satisfying certain conditions.

An order attribute can be used to specify what ordering is to be used. When the nature of the child elements permits, the ordering defaults to a numeric or lexicographic ordering.

Sets are structured much the same as lists except that there is no implied ordering and the type of set may be "normal" or "multiset" with "multiset" indicating that repetitions are allowed.

For both sets and lists, the child elements must be valid MathML content elements. The type of the child elements is not restricted. For example, one might construct a list of equations, or of inequalities.

Classification

constructor

MMLattribute

Name	Value	Default
order	lexicographic | numeric	none

Signature

(anything*) -> list

[order=ordering](anything*) -> list

(domainofapp,function) -> list

(bvar+,domainofapp,anything) -> list

[order=ordering](domainofapp,function) -> list(ordering)

[order=ordering](bvar*,domainofapp,anything) -> list(ordering)

Example

<list><ci>a</ci><ci>b</ci><ci>c</ci></list>

Example

<list order="numeric">
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><lt/><ci>x</ci><cn>5</cn></apply>
  </condition>
</list>

C.2 MMLdefinition: list

Description

The list element is the container element that constructs a list of elements. They may be explicitly listed, or defined by expressions or functions evaluated over a domain of application. The domain of application may be given explicitly, or provided by means of one of the shortcut notations.

Classification

constructor

MMLattribute

Name	Value	Default
order	lexicographic | numeric	none

Signature

(anything*) -> list

[order=ordering](anything*) -> list

(domainofapp,function) -> list

(bvar+,domainofapp,anything) -> list

[order=ordering](domainofapp,function) -> list(ordering)

[order=ordering](bvar*,domainofapp,anything) -> list(ordering)

Example

<list>
  <ci> a </ci>
  <ci> b </ci>
  <ci> c </ci>
</list>

Example

<list order="numeric">
  <bvar><ci> x </ci></bvar>
  <condition>
    <apply><lt/>
      <ci> x </ci>
      <cn> 5 </cn>
    </apply>
  </condition>
</list>

C.3 MMLdefinition: union

Description

This is the set-theoretic operation of union of sets. This n-ary operator generalizes to operations on multisets by tracking the frequency of occurrence of each element in the union. As an n-ary operation the operands can be generated by allowing a function or expression to range over the elements of a domain of application. Thus it accepts qualifiers.

Discussion

The union element is the operator for a set-theoretic union or join of two (or more) sets. The operands are usually listed explicitly.

As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

nary set function

Signature

(set*) -> set

(multiset*) -> multiset

(domainofapp, set_valued_function) -> set

(bvar+,domainofapp,set_valued_expression) -> set

(domainofapp, multiset_valued_function) -> multiset

(bvar+,domainofapp,multiset_valued_expression) -> multiset

Example

<apply><union/>
  <ci> A </ci>
  <ci> B </ci>
</apply>

Example

		<apply>
		  <union/>
		  <bvar><ci type="set"> S </ci></bvar>
		  <domainofapplication><ci type="list">L</ci></domainofapplication>
		  <ci type="set"> S </ci>
		</apply>
		

C.4 MMLdefinition: intersect

Description

This n-ary operator indicates the intersection of two sets. If the two sets are multisets, the result is a multiset. in which each element is present with a repetition determined by the smallest number of occurrences in any of the sets (multisets) that occur as arguments.

Discussion

The intersect element is the operator for the set-theoretic intersection or meet of two (or more) sets. The operands are usually listed explicitly.

As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

nary function

Signature

(set*) -> set

(multiset*) -> multiset

(domainofapp, set_valued_function) -> set

(bvar+,domainofapp,set_valued_expression) -> set

(domainofapp, multiset_valued_function) -> multiset

(bvar+,domainofapp,multiset_valued_expression) -> multiset

Example

<apply><intersect/>
  <ci type="set"> A </ci>
  <ci type="set"> B </ci>
</apply>

Example

<apply>
  <intersect/>
  <bvar><ci type="set"> S </ci></bvar>
  <domainofapplication><ci type="list">L</ci></domainofapplication>
  <ci type="set"> S </ci>
</apply>

C.5 MMLdefinition: in

Description

The in element is the relational operator used for a set-theoretic inclusion ('is in' or 'is a member of').

Discussion

The in element is the relational operator used for a set-theoretic inclusion ("is in" or "is a member of").

The in element is a binary set relation.

Classification

function

Signature

(anything,set) -> boolean

(anything,multiset) -> boolean

Example

<apply><in/>
  <ci> a </ci>
  <ci type="set"> A </ci>
</apply>

C.6 MMLdefinition: notin

Description

The notin element is the relational operator element used to construct set-theoretic exclusion ('is not in' or 'is not a member of').

Discussion

The notin element is the relational operator element used for set-theoretic exclusion ("is not in" or "is not a member of").

The notin element is a binary set relation.

Classification

function

Signature

(anything,set) -> boolean

(anything,multiset) -> boolean

Example

<apply><notin/>
  <ci> a </ci>
  <ci type="set"> A </ci>
</apply>

C.7 MMLdefinition: subset

Description

The subset element is the n-ary relational operator element for a set-theoretic containment ('is a subset of').

Discussion

The subset element is the relational operator element for a set-theoretic containment ("is a subset of").

The subset element is an n-ary set relation. As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

Signature

(set*) -> boolean

(multiset*) -> boolean

(domainofapp,function) -> boolean

(bvar+,domainofapp,algebraic) -> boolean

Example

<apply><subset/>
  <ci type="set"> A </ci>
  <ci type="set"> B </ci>
</apply>

Example

<apply>
  <subset/>
  <subset/>
  <bvar><ci type="set">S</ci></bvar>
  <condition>
    <apply><in/>
      <ci>S</ci>
      <ci type="list">T</ci>
    </apply>
  </condition>
  <ci>S</ci>
</apply>

C.8 MMLdefinition: prsubset

Description

The prsubset element is the n-ary relational operator element for set-theoretic proper containment ('is a proper subset of').

Discussion

The prsubset element is the relational operator element for set-theoretic proper containment ("is a proper subset of").

The prsubset element is an n-ary set relation. As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

function

Signature

(set*) -> boolean

(multiset*) -> boolean

(domainofapp,function) -> boolean

(bvar+,domainofapp,boolean) -> boolean

Example

<apply><prsubset/>
  <ci type="set"> A </ci>
  <ci type="set"> B </ci>
</apply>

Example

<apply>
  <prsubset/>
  <bvar><ci type="integer">i</ci></bvar>
  <lowlimit><cn>0</cn></lowlimit>
  <uplimit><cn>10</cn></uplimit>
  <apply><selector/>
    <ci type="vector_of_sets">S</ci>
    <ci>i</ci>
  </apply>
</apply>

C.9 MMLdefinition: notsubset

Description

The notsubset element is the relational operator element for the set-theoretic relation 'is not a subset of'.

Discussion

The notsubset element is the relational operator element for the set-theoretic relation "is not a subset of".

The notsubset element is a binary set relation.

Classification

function

Signature

(set, set) -> boolean

(multiset,multiset) -> boolean

Example

<apply><notsubset/>
  <ci type="set"> A </ci>
  <ci type="set"> B </ci>
</apply>

C.10 MMLdefinition: notprsubset

Description

The notprsubset element is the element for constructing the set-theoretic relation 'is not a proper subset of'.

Discussion

The notprsubset element is the operator element for the set-theoretic relation "is not a proper subset of".

The notprsubset element is a binary set relation.

Classification

function

Signature

(set,set) -> boolean

(multiset,multiset) -> boolean

Example

<apply><notprsubset/>
  <ci type="set"> A </ci>
  <ci type="set"> B </ci>
</apply>

C.11 MMLdefinition: setdiff

Description

The setdiff element is the operator element for a set-theoretic difference of two sets.

Discussion

The setdiff element is the operator element for a set-theoretic difference of two sets.

Classification

binary set function

Signature

(set,set) -> set

(multiset,multiset) -> multiset

Example

<apply><setdiff/>
  <ci type="set"> A </ci>
  <ci type="set"> B </ci>
</apply>

C.12 MMLdefinition: card

Description

The card element is the operator element for deriving the size or cardinality of a set. The size of a multset is simply the total number of elements in the multiset.

Discussion

The card element is the operator element for the size or cardinality of a set.

Classification

unary function

Signature

(set) -> scalar

(multiset) -> scalar

Example

where A is a set with 5 elements.

		
		<apply><eq/>
  <apply><card/><ci> A </ci></apply>
  <ci> 5 </ci>
</apply>

C.13 MMLdefinition: cartesianproduct

Description

The cartesianproduct element is the operator for a set-theoretic cartesian product of two (or more) sets. The cartesian product of multisets produces a multiset since n-tuples may be repeated if elements in the base sets are repeated.

Discussion

The cartesianproduct element is the operator element for the Cartesian product of two or more sets. If A and B are two sets, then the Cartesian product of A and B is the set of all pairs (a,b) with a in A and b in B.

As an n-ary operator, its operands may also be generated as described in Section 4.4.10 Lifted Associative Commutative Operators Therefore it may take qualifiers.

Classification

nary function

Signature

(set*) -> set

(multiset*) -> multiset

Example

<apply><cartesianproduct/>
  <ci> A </ci>
  <ci> B </ci>
</apply>

Example

<apply><cartesianproduct/>
  <reals/>
  <reals/>
  <reals/>
</apply>

3.8 Content Dictionary: Sequences and Series

C.1 MMLdefinition: sum

Description

The sum element denotes the summation operator. It may be qualified by providing a domainofapplication. This may be provided using one of the shorthand notations for domainofapplication such as an uplimit,lowlimit pair or a condition or an interval. The index for the summation is specified by a bvar element.

Discussion

The sum element denotes the summation operator. The most general form of a sum specifies the terms of the sum by using a domainofapplication element to specify a domain. If no bound variables are specified then terms of the sum correspond to those produced by evaluating the function that is provided at the points of the domain, while if bound variables are present they are the index of summation and they take on the values of points in the domain. In this case the terms of the sum correspond to the values of the expression that is provided, evaluated at those points. Depending on the structure of the domain, the domain of summation can be abbreviated by using uplimit and lowlimit to specify upper and lower limits for the sum.

The sum function accept the bvar, lowlimit, uplimit, interval, condition and domainofapplication schemata. If both lowlimit and uplimit schemata are present, they denote the limits of the sum or product. The limits may alternatively be specified using the interval, condition or domainofapplication schema. The bvar schema signifies the internal variable in the sum or product. A typical example might be:

<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>

When used with sum or product, each qualifier schema is expected to contain a single child schema; otherwise an error is generated.

Classification

function

Signature

(function ) -> function

(bvar,algebraic ) -> algebraic

(domainofapp,function) -> function

(bvar+,domainofapp,algebraic) -> algebraic

Example

<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>

Example

<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>

Example

	  <apply>
  <sum/>
  <domainofapplication>
    <ci type="set"> B </ci>
  </domainofapplication>
  <ci type="function"> f </ci>
</apply>

C.2 MMLdefinition: product

Description

The product element denotes the product operator. It may be qualified by providing a domainofapplication. This may be provided using one of the shorthand notations for domainofapplication such as an uplimit,lowlimit pair or a condition or an interval. The index for the product is specified by a bvar element.

Discussion

The product element denotes the product operator. The most general form of a product specifies the terms of the product by using a domainofapplication element to specify the domain. If no bound variables are specified then terms of the product correspond to those produced by evaluating the function that is provided at the points of the domain, while if bound variables are present they are the index of product and they take on the values of points in the domain. In this case the terms of the product correspond to the values of the expression that is provided, evaluated at those points. Depending on the structure of the domain, the domain of product can be abbreviated by using uplimit and lowlimit to specify upper and lower limits for the product.

The sum function accepts the bvar, lowlimit, uplimit, interval, condition and domainofapplication schemata. If both lowlimit and uplimit schemata are present, they denote the limits of the sum or product. The limits may alternatively be specified using the interval, condition or domainofapplication schema. The bvar schema signifies the internal variable in the sum or product.

Classification

function

Signature

(function) -> function

(bvar,algebraic) -> algebraic

(domainofapp,function) -> function

(bvar+,domainofapp,algebraic) -> algebraic

Example

<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>

Example

<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>

C.3 MMLdefinition: limit

Description

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.

Discussion

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.

The limit function accepts zero or more bvar schemata, and optional condition and lowlimit schemata. A condition may be used to place constraints on the bvar. The bvar schema denotes the variable with respect to which the limit is being taken. The lowlimit schema denotes the limit point. When used with limit, the bvar and lowlimit schemata are expected to contain a single child schema; otherwise an error is generated.

Classification

function

Signature

(bvar+, lowlimit, uplimit, algebraic) -> real

(bvar+, condition , algebraic) -> real

Example

<apply>
  <limit/>
  <bvar><ci> x </ci></bvar>
  <lowlimit><cn> 0 </cn></lowlimit>
  <apply><sin/><ci> x </ci></apply>
</apply>

Example

<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>

Example

		<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>
		

C.4 MMLdefinition: tendsto

Description

The tendsto element is used to express the relation that a quantity is tending to a specified value.

Discussion

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 that are not explicitly covered by the recommendation.

The tendsto element takes the attributes type to set the direction from which the limiting value is approached. It occurs as a container so that it can more easily be used in the construction of a limit expression. Predefined values: "above", "below", "two-sided". The default value is "two-sided".

Issue type_on_tendsto	wiki (member only)
keep type attribute on tendsto?
It seems very awkward to keep this, it would probably be better to have an tendsto_xxx symbol where xxx is the typevalue. (this would be closer to what OpenMath does now).
Resolution	None recorded

Classification

function

MMLattribute

Name	Value	Default
type	above | below | all | MathMLType	all

Signature

(algebraic,algebraic) -> tendsto

[ type=direction ](algebraic,algebraic) -> tendsto(direction)

Example

		
<apply><tendsto type="above"/>
  <apply><power/><ci> x </ci><cn> 2 </cn></apply>
  <apply><power/><ci> a </ci><cn> 2 </cn></apply>
</apply>

Example

To express (x, y) (f(x, y), g(x,y)), one might use vectors, as in:

		
		
<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>

3.9 Content Dictionary: Elementary Classical Functions

The names of the common trigonometric functions supported by MathML are listed below. Since their standard interpretations are widely known, they are discussed as a group.

sin	cos	tan
sec	csc	cot
sinh	cosh	tanh
sech	csch	coth
arcsin	arccos	arctan
arccosh	arccot	arccoth
arccsc	arccsch	arcsec
arcsech	arcsinh	arctanh

Discussion

These operator elements denote the standard trigonometric functions.

C.1 MMLdefinition: exp

Description

This element represents the exponentiation function as described in Abramowitz and Stegun, section 4.2. It takes one argument.

Discussion

The exp element represents the exponential function associated with the inverse of the ln function. In particular, exp(1) is approximately 2.718281828.

Classification

unary function

Signature

(real) -> real

(complex) -> complex

Property

<apply><eq/>
  <apply><exp/><cn>0</cn></apply>
  <cn>1</cn>
</apply>

Property

for all k if k is an integer then e^(z+2*pi*k*i)=e^z

Example

<apply><exp/><ci> x </ci></apply>

C.2 MMLdefinition: ln

Description

This element represents the ln function (natural logarithm) as described in Abramowitz and Stegun, section 4.1. It takes one argument.

Discussion

The ln element represents the natural logarithm function.

Classification

unary function

Signature

(real) -> real

(complex) -> complex

Property

-pi lt Im ln x leq pi

Example

If a = e, (where e is the base of the natural logarithms) this will yield the value 1.

		
		<apply><ln/><ci> a </ci></apply>

C.3 MMLdefinition: log

Description

This element represents the log function. It is defined in Abramowitz and Stegun, Handbook of Mathematical Functions, section 4.1 If its first argument is a logbase element, it specifies the base and the second argument is the argument to which the function is applied using that base. If no logbase element is present, the base is assumed to be 10.

Discussion

The log element is the operator that returns a logarithm to a given base. The base may be specified using a logbase element, which should be the first element following log, i.e. the second child of the containing apply element. If the logbase element is not present, a default base of 10 is assumed.

The log function accepts only the logbase schema. If present, the logbase schema denotes the base with respect to which the logarithm is being taken. Otherwise, the log is assumed to be base 10. When used with log, the logbase schema is expected to contain a single child schema; otherwise an error is generated.

Classification

function

Signature

(logbase,real) -> real

(logbase,complex) -> complex

(real) -> real

(complex) -> complex

Property

a^b = c implies log_a c = b

Example

This markup represents "the base 3 logarithm of x". For natural logarithms base e, the ln element should be used instead.

		
		<apply><log/>
  <logbase><cn> 3 </cn></logbase>
  <ci> x </ci>
</apply>

Example

<apply><log/><ci>x</ci></apply>

C.4 MMLdefinition: sin

Description

This element represents the sin function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

sin(0) = 0

Property

sin(integer*Pi) = 0

Property

sin(x) = (exp(ix)-exp(-ix))/2i

Example

<apply><sin/><ci> x </ci></apply>

Example

<apply>
  <sin/>
  <apply>
    <plus/>
    <apply><cos/>
      <ci> x </ci>
    </apply>
    <apply>
      <power/>
      <ci> x </ci>
      <cn> 3 </cn>
    </apply>
  </apply>
</apply>

C.5 MMLdefinition: cos

Description

This element represents the cos function as described in Abramowitz and Stegun, section 4.3. It takes one argument. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

cos(0) = 1

Property

cos(integer*Pi+Pi/2) = 0

Property

cos(x) = (exp(ix)+exp(-ix))/2

Example

<apply><cos/><ci>x</ci></apply>

C.6 MMLdefinition: tan

Description

This element represents the tan function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

tan(integer*Pi) = 0

Property

tan(x) = sin(x)/cos(x)

Example

<apply><tan/><ci>x</ci></apply>

C.7 MMLdefinition: sec

Description

This element represents the sec function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

sec(x) = 1/cos(x)

Example

<apply><sec/><ci>x</ci></apply>

C.8 MMLdefinition: csc

Description

This element represents the csc function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

csc(x) = 1/sin(x)

Example

<apply><csc/><ci>x</ci></apply>

C.9 MMLdefinition: cot

Description

This element represents the cot function as described in Abramowitz and Stegun, section 4.3. It takes one argument. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

cot(integer*Pi+Pi/2) = 0

Property

cot(x) = cos(x)/sin(x)

Property

cot A = 1/tan A

Example

<apply><cot/><ci>x</ci></apply>

C.10 MMLdefinition: sinh

Description

This element represents the sinh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

sinh A = 1/2 * (e^A - e^(-A))

Example

<apply><sinh/><ci>x</ci></apply>

C.11 MMLdefinition: cosh

Description

This symbol represents the cosh function as described in Abramowitz and Stegun, section 4.5. It takes one argument. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

cosh A = 1/2 * (e^A + e^(-A))

Example

<apply><cosh/><ci>x</ci></apply>

C.12 MMLdefinition: tanh

Description

This element represents the tanh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

tanh A = sinh A / cosh A

Example

<apply><tanh/><ci>x</ci></apply>

C.13 MMLdefinition: sech

Description

This element represents the sech function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

sech A = 1/cosh A

Example

<apply><sech/><ci>x</ci></apply>

C.14 MMLdefinition: csch

Description

This element represents the csch function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

csch A = 1/sinh A

Example

<apply><csch/><ci>x</ci></apply>

C.15 MMLdefinition: coth

Description

This element represents the coth function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

coth A = 1/tanh A

Example

<apply><coth/><ci>x</ci></apply>

C.16 MMLdefinition: arcsin

Description

This element represents the arcsin function which is the inverse of the sin function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arcsin(z) = -i ln (sqrt(1-z^2)+iz)

Example

<apply><arcsin/><ci>x</ci></apply>

C.17 MMLdefinition: arccos

Description

This element represents the arccos function which is the inverse of the cos function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arccos(z) = -i ln(z+i \sqrt(1-z^2))

Example

<apply><arccos/><ci>x</ci></apply>

C.18 MMLdefinition: arctan

Description

This element represents the arctan function which is the inverse of the tan function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arctan(z) = (ln(1+iz)-ln(1-iz))/2i

Example

<apply><arctan/><ci>x</ci></apply>

C.19 MMLdefinition: arccosh

Description

This symbol represents the arccosh function as described in Abramowitz and Stegun, section 4.6. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arccosh(z) = 2*ln(\sqrt((z+1)/2) + \sqrt((z-1)/2))

Example

<apply><arccosh/><ci>x</ci></apply>

C.20 MMLdefinition: arccot

Description

This element represents the arccot function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arccot(-z) = - arccot(z)

Example

<apply><arccot/><ci>x</ci></apply>

C.21 MMLdefinition: arccoth

Description

This element represents the arccoth function as described in Abramowitz and Stegun, section 4.6. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arccoth(z) = (ln(-1-z)-ln(1-z))/2

Example

<apply><arccoth/><ci>x</ci></apply>

C.22 MMLdefinition: arccsc

Description

This element represents the arccsc function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arccsc(z) = -i ln(i/z + \sqrt(1 - 1/z^2))

Example

<apply><arccsc/><ci>x</ci></apply>

C.23 MMLdefinition: arccsch

Description

This element represents the arccsch function as described in Abramowitz and Stegun, section 4.6. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arccsch(z) = ln(1/z + \sqrt(1+(1/z)^2))

Example

<apply><arccsch/><ci>x</ci></apply>

C.24 MMLdefinition: arcsec

Description

This element represents the arcsec function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arcsec(z) = -i ln(1/z + i \sqrt(1-1/z^2))

Example

<apply><arcsec/><ci>x</ci></apply>

C.25 MMLdefinition: arcsech

Description

This element represents the arcsech function as described in Abramowitz and Stegun, section 4.6. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arcsech(z) = 2 ln(\sqrt((1+z)/(2z)) + \sqrt((1-z)/(2z)))

Example

<apply><arcsech/><ci>x</ci></apply>

C.26 MMLdefinition: arcsinh

Description

This element represents the arcsinh function as described in Abramowitz and Stegun, section 4.6. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arcsinh z = ln(z + \sqrt(1+z^2))

Example

<apply><arcsinh/><ci>x</ci></apply>

C.27 MMLdefinition: arctanh

Description

This element represents the arctanh function as described in Abramowitz and Stegun, section 4.6. It takes one argument.

Classification

function

Signature

(real) -> real

(complex) -> complex

Property

arctanh(z) = - i * arctan(i * z)

Example

<apply><arctanh/><ci>x</ci></apply>

3.10 Content Dictionary: Statistics

C.1 MMLdefinition: mean

Description

The mean value of a set of data, or of a random variable. See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, section 7.7.1

Discussion

mean is the operator element representing a mean or average.

Classification

nary function

Signature

(random_variable) -> scalar

(scalar+) -> scalar

Example

<apply><mean/><ci type="discrete_random_variable"> X </ci></apply>

Example

<apply><mean/><cn>3</cn><cn>4</cn><cn>3</cn><cn>7</cn><cn>4</cn></apply>

Example

<apply><mean/><ci> X </ci></apply>

C.2 MMLdefinition: sdev

Description

This element represents a function denoting the sample standard deviation of its arguments. The arguments are either all data, or a discrete random variable, or a continuous random variable.

For numeric data at least two values are required and this is the square root of (the sum of the squares of the deviations from the mean of the arguments, divided by the number of arguments less one). For a "discrete_random_variable", this is the square root of the second moment about the mean. This further generalizes to identifiers of type continuous_random_variable.

See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, (7.7.11) section 7.7.1.

Discussion

sdev is the operator element representing the statistical standard deviation operator.

Classification

nary function

Signature

(scalar,scalar+) -> scalar

(discrete_random_variable) -> scalar

(continuous_random_variable) -> scalar

Example

<apply><sdev/><cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn></apply>

Example

sdev is an n-ary operator.

      
    <apply><sdev/><ci type="discrete_random_variable"> X </ci></apply>

C.3 MMLdefinition: variance

Description

This symbol represents a function denoting the variance of its arguments, that is, the square of the standard deviation. The arguments are either all data in which case there are two or more of them, or an identifier of type discrete_random_variable, or continuous_random_variable. See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, [7.1.2] and [7.7].

Discussion

variance is the operator element representing the statistical variance operator.

Classification

nary function

Signature

(scalar,scalar+) -> scalar

(descrete_random_variable) -> scalar

(continuous_random_variable) -> scalar

Example

<apply><variance/><cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn></apply>

Example

variance is an n-ary operator.

	  
	  <apply><variance/><ci type="discrete_random_variable"> X </ci></apply>

C.4 MMLdefinition: median

Description

This symbol represents an n-ary function denoting the median of its arguments. That is, if the data were placed in ascending order then it denotes the middle one (in the case of an odd amount of data) or the average of the middle two (in the case of an even amount of data). See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, section 7.7.1

Discussion

median is the operator element representing the statistical median operator.

Classification

nary function

Signature

(scalar+) -> scalar

Example

<apply><median/><cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn></apply>

Example

median is an n-ary operator.

	  
		
		<apply>
		  <median/>
		  <ci> X </ci>
		</apply>
		

C.5 MMLdefinition: mode

Description

This represents the mode of n data values. The mode is the data value that occurs with the greatest frequency. See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, section 7.7.1

Discussion

mode is the operator element representing the statistical mode operator.

Classification

nary function

Signature

(scalar+) -> scalar

Example

<apply><mode/><cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn></apply>

Example

mode is an n-ary operator.

	  
	  <apply>
  <mode/>
  <ci> X </ci>
</apply>

C.6 MMLdefinition: moment

Description

This symbol is used to denote the i'th moment of a set of data, or a random variable. Unless otherwise specified, the moment is about the origin. For example, the i'th moment of X about the origin is given by moment( i , 0 , x ).

The first argument indicates which moment about that point is being specified. For the i'th moment the first argument should be i. The second argument specifies the point about which the moment is computed. It is either an actual point ( e.g. 0 ), or a function which can be used on the data to compute that point. To indicate a central moment, specify the element "mean". The third argument is either a discrete or continuous random variable, or the start of a sequence of data. If there is a sequence of data then the i'th moment is (1/n) (x_1^i + x_2^i + ... + x_n^i).

See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, section 7.7.1

Discussion

The moment element represents the statistical moment operator. Use the qualifier degree for the n in " n-th moment". Use the qualifier momentabout for the p in "moment about p".

The moment function accepts the degree and momentabout schema. 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 schema; otherwise an error is generated. 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.

Classification

function

Signature

(degree,momentabout?,scalar+) -> scalar

(degree,momentabout?,discrete_random_variable) -> scalar

(degree,momentabout?,continuous_random_variable) -> scalar

Example

The third moment about the point p of a discrete random variable

moment is an operator taking qualifiers (see Section 4.2.7 Qualifiers). The third moment of the distribution X about the point p is written:

          <apply> <moment/>
  <degree><cn>3</cn></degree>
  <momentabout><ci>p</ci></momentabout>
  <ci>X</ci>
</apply>

Example

The 3rd central moment of a set of data.

          <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>

Example

The 3rd central moment of a discrete random variable.

          <apply><moment/>
  <degree><cn>3</cn></degree>
  <momentabout><mean/></momentabout>
  <ci type="discrete_random_variable"> X </ci>
</apply>

Example

The 3rd moment about the origin of a set of data.

          <apply><moment/>
  <degree><cn>3</cn></degree>
  <momentabout><cn>0</cn></momentabout>
  <cn>6</cn><cn>4</cn><cn>2</cn><cn>2</cn>
</apply>

C.7 MMLdefinition: momentabout

Description

This qualifier element is used to identify the point about which a moment is to be computed. It may be an explicit point, or it may identify a method by which the point is to be computed from the given data. For example the moment may be computed about the mean by specifying the element used for the mean.

Discussion

The momentabout element is a qualifier element used with the moment element to represent statistical moments. Use the qualifier momentabout for the p in "moment about p".

Classification

constructor

Signature

(function) -> method

(scalar) -> point

Example

The third moment of the distribution X about the point p of a discrete random variable is written:

          
<apply> <moment/>
  <degree> <cn> 3 </cn> </degree>
  <momentabout> <ci> p </ci> </momentabout>
  <ci> X </ci>
</apply>

Example

The 3rd central moment of a set of data.

          
<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>

3.11 Content Dictionary: Linear Algebra

C.1 MMLdefinition: vector

Description

A vector is an ordered n-tuple of values representing an element of an n-dimensional vector space. The "values" are all from the same ring, typically real or complex. Where orientation is important, such as for pre or post multiplication by a matrix a vector is treated as if it were a column vector and its transpose is treated a row vector. The type attribute can be used to explicitly specify that a vector is a "row" vector. See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, [2.4]

Discussion

vector is the container element for a vector. The child elements form the components of the vector.

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 behaves the same as a matrix consisting of a single row. Note that vectors may be rendered either as a single column or row.

In general a vector can be constructed by providing a function and a 1-dimensional domain of application. The entries of the vector correspond to the values obtained by evaluating the function at the points of the domain. The qualifications defined by a domainofapplication element can also be abbreviated in several ways including a condition placed on a bound variable and an expression involving that variable.

The vector element constructs vectors from an n-dimensional vector space so that its n child elements typically represent real or complex valued scalars as in the three-element vector

<vector>
  <apply><plus/><ci>x</ci><ci>y</ci></apply>
  <cn>3</cn>
  <cn>7</cn>
</vector>

A vector can also be constructed by evaluating a function over a specific domain of application, each evaluation corresponding to an entry in the vector. In its most general form a domain is explicitly specified by a domainofapplication element and a function. Optionally the domainofapplication can be augmented by a bvar element and an algebraic expression expressed in terms of it. Qualifications defined by a domainofapplication element can be abbreviated in several ways as described in Section 4.2.7 Qualifiers.

Classification

constructor

MMLattribute

Name	Value	Default
type	row | column | MathMLType	column

Signature

(real*) -> vector(type=real)

[type=vectortype]((anything)*) -> vector(type=vectortype)

(domainofapp,function) -> vector

(bvar+,domainofapp,anything) -> vector

Property

vector=column_vector

Property

matrix * vector = vector

Property

matrix * column_vector = column_ vector

Property

row_vector*matrix = row_vector

Property

transpose(vector) = row_vector

Property

transpose(column_vector) = row_vector

Property

transpose(row_vector) = column_vector

Property

distributive over scalars

Property

associativity.

Property

Matrix * column vector

Property

row vector * Matrix

Example

vector is a constructor element.

	  
	  <vector><cn>1</cn><cn>2</cn><cn>3</cn><ci>x</ci></vector>
	

Example

	  <vector type="row"><cn>1</cn><cn>2</cn><cn>3</cn><ci>x</ci></vector>
	

Example

<vector>
  <bvar><ci type="integer">i</ci></bvar>
  <lowlimit><ci>1</ci></lowlimit>
  <uplimit><ci>10</ci></uplimit>
  <apply><power/>
    <ci>x</ci>
    <ci>i</ci>
  </apply>
</vector>

C.2 MMLdefinition: matrix

Description

This is the constructor for a matrix. It requires matrixrow's as arguments. It is used to represent matrices. See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, [2.5.1].

Discussion

The matrix element is the container element for matrix rows, which are represented by matrixrow. The matrixrows contain the elements of a matrix.

In general a matrix can be constructed 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. The qualifications defined by a domainofapplication element can also be abbreviated in several ways including a condition element placing constraints directly on bound variables and an expression in those variables.

The matrix element is used to represent mathematical matrices. It has zero or more child elements, all of which are matrixrow elements. These in turn expect zero or more child elements that evaluate to algebraic expressions or numbers. These sub-elements are often real numbers, or symbols as in

<matrix>
  <matrixrow> <cn> 1 </cn> <cn> 2 </cn> </matrixrow>
  <matrixrow> <cn> 3 </cn> <cn> 4 </cn> </matrixrow>
</matrix>

The matrixrow elements must always be contained inside of a matrix, and all rows in a given matrix must have the same number of elements.

Note that the behavior of the matrix and matrixrow elements is substantially different from the mtable and mtr presentation elements.

A matrix can also be constructed by evaluating a bivariate function over a specific domain of application, each evaluation corresponding to an entry in the matrix. In its most general form a domain of application is explicitly specified by a domainofapplication element and a function which when evaluated at points of the domain produces entries in the matrix. Optionally the domainofapplication can be augmented by bvar elements and an algebraic expression expressed in terms of them.

Classification

constructor

MMLattribute

Name	Value	Default
type	real | complex | integer | symbolic | integer | MathMLType	real

Signature

(matrixrow*) -> matrix

[type=matrixtype](matrixrow*) -> matrix(type=matrixtype)

(domainofapp,function) -> matrix

(bvar,bvar,domainofapp,anything) -> matrix

Property

scalar multiplication

Property

scalar multiplication

	  
	  Matrix*column vector
	

Property

scalar multiplication

          
	  Addition
	

Property

scalar multiplication

          
	  Matrix*Matrix
	

Example

matrix is a constructor element.

	  
	  <matrix>
  <matrixrow><cn>0</cn> <cn> 1 </cn> <cn> 0 </cn></matrixrow>
  <matrixrow><cn>0</cn> <cn> 0 </cn> <cn> 1 </cn></matrixrow>
  <matrixrow><cn>1</cn> <cn> 0 </cn> <cn> 0 </cn></matrixrow>
</matrix>

Example

<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>
</vector>

C.3 MMLdefinition: matrixrow

Description

This symbol is an n-ary constructor used to represent rows of matrices. Its arguments should be members of a ring.

Discussion

The matrixrow element is the container element for the rows of a matrix.

Classification

constructor

Signature

(ringelement+) -> matrixrow

Example

	  <matrixrow><cn>1</cn><cn>2</cn></matrixrow>
	

C.4 MMLdefinition: determinant

Description

The "determinant" of a matrix. This is a unary function. See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, [2.5.4].

Discussion

The determinant element is the operator for constructing the determinant of a matrix.

Classification

function

Signature

(matrix)-> scalar

Example

determinant is a unary operator.

	  
	  <apply><determinant/>
  <ci type="matrix"> A </ci>
</apply>

C.5 MMLdefinition: transpose

Description

The transpose of a matrix or vector. See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, [2.4] and [2.5.1].

Discussion

The transpose element is the operator for constructing the transpose of a matrix.

Classification

function

Signature

(vector)->vector(type=row)

(matrix)->matrix

(vector[type=row])->vector

Property

transpose(transpose(A))= A

Property

transpose(transpose(V))= V

Example

transpose is a unary operator.

	  
	  <apply><transpose/><ci type="matrix">A</ci></apply>
	

Example

<apply><transpose/><ci type="vector">V</ci></apply>

C.6 MMLdefinition: selector

Description

The operator used to extract sub-objects from vectors, matrices matrix rows and lists. Elements are accessed by providing one index element for each dimension. For matrices, sub-matrices are selected by providing one fewer index items. For a matrix A and a column vector V : select(i, j, A) is the i,j th element of A. select(i, A) is the matrixrow formed from the i'th row of A. select(i, V) is the i'th element of V. select(V) is the sequence of all elements of V. select(A) is the sequence of all elements of A, extracted row by row. select(i, L) is the i'th element of a list. select(L) is the sequence of elements of a list.

Discussion

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. That is, for a matrix ( a_i,j), where the indices denote row and column, the ordering would be a _1,1, a _1,2, ... a _2,1, a_2,2 ... etc.

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 of the selected element.

When two arguments are given, and the first is a vector or list, the second argument specifies an element in the list or vector. When a matrix and only one index i is specified as in

Example

<apply>
  <selector/>
  <matrix>
    <matrixrow><cn>1</cn><cn>2</cn></matrixrow>
    <matrixrow><cn>3</cn><cn>4</cn></matrixrow>
  </matrix>
  <cn>1</cn>
</apply>

it refers to the i-th matrixrow. Thus, the preceding example selects the following row:

Example

<matrixrow> <cn> 1 </cn> <cn> 2 </cn> </matrixrow>

selector is classified as an n-ary linear algebra operator even though it can take only one, two, or three arguments.

Classification

function

Signature

(matrix,scalar,scalar)->scalar

(matrix,scalar)->matrixrow

(matrix)->scalar*

((vector|list|matrixrow),scalar )->scalar

(vector|list|matrixrow)->scalar*

Property

For all vectors V, V = vector(selector(V))

Property

For all matrix rows Mrow, Mrow = matrixrow(selector(Mrow))

Example

<apply><selector/><ci type="matrix">M</ci><cn>3</cn><cn>2</cn></apply>

C.7 MMLdefinition: vectorproduct

Description

The vector or cross product of two nonzero three-dimensional vectors v1 and v2 is defined by

v1 x v2 = n norm(v1) * norm(v2) sin(theta) where n is the unit normal vector perpendicular to both, adhering to the right hand rule. CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, [2.4]

Discussion

The vectorproduct is the operator element for deriving the vector product of two vectors.

The vectorproduct element is a binary vector operator.

Classification

function

Signature

(vector,vector)->vector

Property

if v1 and v2 are parallel then their vector product is 0

Example

<apply><vectorproduct/><ci>u</ci><ci>v</ci></apply>

Example

where A and B are vectors, N is a unit vector orthogonal to A and B, a, b are the magnitudes of A, B and is the angle between A and B.

		
		
		<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> &theta; </ci>
		    </apply>
		    <ci type="vector"> N </ci>
		  </apply>
		</apply>
		

C.8 MMLdefinition: scalarproduct

Description

This symbol represents the scalar product function. It takes two vector arguments and returns a scalar value. The scalar product of two vectors a, b is defined as |a| * |b| * cos(\theta), where \theta is the angle between the two vectors and |.| is a euclidean size function. Note that the scalar product is often referred to as the dot product.

Discussion

The scalarproduct is the operator element for deriving the scalar product of two vectors.

The scalarproduct element is a binary vector operator.

Classification

function

Signature

(vector,vector) -> scalar

Property

if the scalar product of two vectors is 0 then they are orthogonal.

Example

<apply><scalarproduct/><ci>u</ci><ci>v</ci></apply>

Example

where A and B are vectors, a, b are the magnitudes of A, B and is the angle between A and B.

		
	<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> &theta; </ci>
    </apply>
  </apply>
</apply>

C.9 MMLdefinition: outerproduct

Description

This symbol represents the outer product function. It takes two vector arguments and returns a matrix. It is defined as follows: if we write the {i,j}'th element of the matrix to be returned as m_{i,j}, then: m_{i,j}=a_i * b_j where a_i,b_j are the i'th and j'th elements of a, b respectively.

Discussion

The outerproduct is the operator element for deriving the outer product of two vectors.

The outerproduct element is a binary vector operator.

Classification

function

Signature

(vector,vector) -> matrix

Example

<apply><outerproduct/><ci>u</ci><ci>v</ci></apply>

Example

where A and B are vectors.

	  
	  <apply>
  <outerproduct/>
  <ci type="vector">A</ci>
  <ci type="vector">B</ci>
</apply>

3.12 Content Dictionary: Constants and Symbol Elements

C.1 MMLdefinition: integers

Description

integers represents the set of all integers.

Classification

constant

Signature

set

Property

n is an integer implies n+1 is an integer.

      <apply><implies/>
  <apply><in/><ci>n</ci><integers/></apply>
  <apply><in/><apply><plus/><ci>n</ci><cn>1</cn></apply><integers/></apply>
</apply>

Property

0 is an integer

          
<apply><in/><cn>0</cn><integers/></apply>
	

Property

n is an integer implies -n is an integer

          
<apply><implies/>
  <apply><in/><ci>n</ci><integers/></apply>
  <apply><in/><apply><minus/><ci>n</ci></apply><integers/></apply>
</apply>

Example

<apply><in/>
  <cn type="integer"> 42 </cn>
  <integers/>
</apply>

C.2 MMLdefinition: reals

Description

reals represents the set of all real numbers.

Discussion

reals represents the set of all real numbers.

Classification

symbol

Signature

set

Property

(S \subset R and exists y in R : forall x in S x \le y) implies exists z in R such that ( ( forall x in S x \le z) and ( (forall x in S x \le w) implies z le w))

Property

for all a,b | a,b rational with a<b implies there exists rational a,c s.t. a<c and c<b

Example

<apply><in/>
  <cn type="real"> 44.997 </cn>
  <reals/>
</apply>

C.3 MMLdefinition: rationals

Description

rationals represents the set of all rational numbers.

Discussion

rationals represents the set of all rational numbers.

Classification

constant

Signature

set

Property

for all z where z is a rational, there exists integers p and q with p/q = z

          
<apply><forall/>
  <bvar><ci>z</ci></bvar>
  <condition><apply><in/><ci>z</ci><rationals/></apply></condition>
  <apply><exists/>
    <bvar><ci>p</ci></bvar>
    <bvar><ci>q</ci></bvar>
    <apply><and/>
      <apply><in/><ci>p</ci><integers/></apply>
      <apply><in/><ci>q</ci><integers/></apply>
      <apply><eq/>
        <apply><divide/><ci>p</ci><ci>q</ci></apply><integers/></apply>
        <ci>z</ci>
      </apply>
    </apply>
  </apply>

Property

ForAll( [a,b], a and b are rational, a < b implies there exists c such that a < c and c < b )

Example

<apply><in/>
  <cn type="rational"> 22 <sep/>7</cn>
  <rationals/>
</apply>

C.4 MMLdefinition: naturalnumbers

Description

naturalnumbers represents the set of all natural numbers, i.e.. non-negative integers.

Discussion

naturalnumbers represents the set of all natural numbers, i.e. non-negative integers.

Classification

constant

Signature

set

Property

For all n | n is a natural number implies n+1 is a natural number.

          
<apply><forall/>
  <bvar><ci>n</ci></bvar>
  <apply><implies/>
    <apply><in/><ci>n</ci><naturalnumbers/></apply>
    <apply><in/>
      <apply><plus/><ci>n</ci><cn>1</cn></apply>
      <naturalnumbers/>
    </apply>
  </apply>
</apply>

Property

0 is a natural number.

          
<apply><in/><cn>0</cn><naturalnumbers/></apply>

Property

for all n | n in the natural numbers is equivalent to saying n=0 or n-1 is a natural number

Example

<apply><in/>
  <cn type="integer">1729</cn>
  <naturalnumbers/>
</apply>

C.5 MMLdefinition: complexes

Description

complexes represents the set of all complex numbers, i.e., numbers which may have a real and an imaginary part.

Discussion

complexes represents the set of all complex numbers, i.e. numbers which may have a real and an imaginary part.

Classification

constant

Signature

set

Property

for all z | if z is complex then there exist reals x,y s.t. z = x + i * y

Example

<apply><in/>
  <cn type="complex">17<sep/>29</cn>
  <complexes/>
</apply>

C.6 MMLdefinition: primes

Description

primes represents the set of all natural prime numbers, i.e., integers greater than 1 which have no positive integer factor other than themselves and 1.

Discussion

primes represents the set of all natural prime numbers, i.e. integers greater than 1 which have no positive integer factor other than themselves and 1.

Classification

constant

Signature

set

Property

ForAll( [d,p], p is prime, Implies( d | p , d=1 or d=p ) )

          
<apply><forall/>
  <bvar><ci>d</ci></bvar>
  <bvar><ci>p</ci></bvar>
  <condition>
    <apply><and/>
    <apply><in/><ci>p</ci><primes/></apply>
    <apply><in/><ci>d</ci><naturalnumbers/></apply>
    </apply>
  </condition>
  <apply><implies/>
    <apply><factorof/><ci>d</ci><ci>p</ci></apply>
    <apply><or/>
      <apply><eq/><ci>d</ci><cn>1</cn></apply>
      <apply><eq/><ci>d</ci><ci>p</ci></apply>
    </apply>
  </apply>
</apply>

Example

<apply>
<in/>
<cn type="integer">17</cn>
<primes/>
</apply>

C.7 MMLdefinition: exponentiale

Description

exponentiale represents the mathematical constant which is the exponential base of the natural logarithms, commonly written e . It is approximately 2.718281828..

Discussion

exponentiale represents the mathematical constant which is the exponential base of the natural logarithms, commonly written e. It is approximately 2.718281828..

Classification

constant

Signature

real constant

Property

ln(e) = 1

          
	  <apply><eq/>
  <apply><ln/><exponentiale/></apply>
  <cn>1</cn>
</apply>

Property

e is approximately 2.718281828

          <apply><approx/>
  <exponentiale/>
  <cn>2.718281828 </cn>
</apply>

Property

e = the sum as j ranges from 0 to infinity of 1/(j!)

Example

<apply> <eq/>
  <apply><ln/><exponentiale/></apply>
  <cn>1</cn>
</apply>

C.8 MMLdefinition: imaginaryi

Description

imaginaryi represents the mathematical constant which is the square root of -1, commonly written i

Discussion

imaginaryi represents the mathematical constant which is the square root of -1, commonly written i.

Classification

constant

Signature

complex

Property

sqrt(-1) = i

          
<apply><eq/>
  <imaginaryi/>
  <apply><root/><degree><cn>2</cn></degree><cn>-1</cn></apply>
</apply>

Example

<apply> <eq/>
  <apply><power/>
    <imaginaryi/>
    <cn>2</cn>
  </apply>
  <cn>-1</cn>
</apply>

C.9 MMLdefinition: notanumber

Description

notanumber represents the result of an ill-defined floating point operation, sometimes also called NaN.

Discussion

notanumber represents the result of an ill-defined floating point operation, sometimes also called NaN.

Classification

constant

Signature

undefined

Example

<apply><eq/>
  <apply><divide/><cn>0</cn><cn>0</cn></apply>
  <notanumber/>
</apply>

C.10 MMLdefinition: true

Description

true represents the logical constant for truth.

Discussion

true represents the logical constant for truth.

Classification

constant

Signature

boolean

Property

not true = false

          
<apply><eq/>
  <apply><not/><true/></apply>
  <false/>
</apply>

Property

For all boolean p, p or true is true

          
<declare type="boolean"><ci>p</ci></declare>
<apply><forall/>
  <bvar><ci>p</ci></bvar>
  <apply><eq/>
    <apply><or/><ci>p</ci><true/></apply>
    <true/>
  </apply>
</apply>

Example

<apply> <eq/>
  <apply><or/>
    <true/>
    <ci type = "boolean">P</ci>
  </apply>
  <true/>
</apply>

C.11 MMLdefinition: false

Description

false represents the logical constant for falsehood.

Discussion

false represents the logical constant for falsehood.

Classification

constant

Signature

boolean

Property

not true = false

          
<apply><eq/>
  <apply><not/><true/></apply>
  <false/>
</apply>

Property

p and false = false

          
<declare type="boolean"><ci>p</ci></declare>
<apply><forall/>
  <bvar><ci>p</ci></bvar>
  <apply><and/><ci>p</ci><false/></apply>
  <false/>
</apply>

Example

<apply><eq/>
  <apply><and/>
    <false/>
    <ci type = "boolean">P</ci>
  </apply>
  <false/>
</apply>

C.12 MMLdefinition: emptyset

Description

emptyset represents the empty set.

Discussion

emptyset represents the empty set.

Classification

constant

Signature

set

Property

for all sets S, intersect(S,emptyset) = emptyset

          
<apply><forall/><bvar><ci type="set">S</ci></bvar>
  <apply><eq/>
    <apply><intersect/><emptyset/><ci>S</ci></apply>
    <emptyset/>
  </apply>
</apply>

Example

<apply><neq/>
  <integers/>
  <emptyset/>
</apply>

C.13 MMLdefinition: pi

Description

pi represents the mathematical constant which is the ratio of a circle's circumference to its diameter, approximately 3.141592653.

Discussion

pi represents the mathematical constant which is the ratio of a circle's circumference to its diameter, approximately 3.141592653.

Classification

constant

Signature

real

Property

<apply><approx/>
  <cn>pi</cn>
  <cn> 3.141592654 </cn>
</apply>

Property

pi = 4 * the sum as j ranges from 0 to infinity of ((1/(4j+1))-(1/(4j+3)))

Example

<apply><approx/>
  <pi/>
  <cn type = "rational">22<sep/>7</cn>
</apply>

C.14 MMLdefinition: eulergamma

Description

A symbol to convey the notion of the gamma constant as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, section 6.1.3. It is the limit of 1 + 1/2 + 1/3 + ... + 1/m - ln m as m tends to infinity, this is approximately 0.5772 15664.

Discussion

eulergamma represents Euler's constant, approximately 0.5772156649

Classification

constant

Signature

real

Property

gamma is approx. 0.5772156649

          
<apply><approx/>
  <eulergamma/>
  <cn> .5772156649 </cn>
</apply>

Property

gamma = limit_(m -> infinity)(sum_(j ranges from 1 to m)(1/j) - ln m)

Example

	  <eulergamma/>

Example

<apply><approx/>
  <eulergamma/>
  <cn>0.5772156649</cn>
</apply>

C.15 MMLdefinition: infinity

Description

Infinity. Interpretation depends on the context. The default value is the positive infinity used to extend the real number line. The "type" attribute can be use to indicate that this is a "complex" infinity.

Discussion

infinity represents the concept of infinity. Proper interpretation depends on context.

Classification

constant

Signature

constant

Property

infinity/infinity is not defined.

          
<apply><eq/>
  <apply><divide/><infinity/><infinity/></apply>
  <notanumber/>
</apply>

Property

for all reals x, x \lt infinity

          
<apply><forall/>
  <bvar><ci>n</ci></bvar>
  <condition><apply><in/><ci>n</ci><reals/></apply></condition>
  <apply><lt/><ci>n</ci><infinity/></apply>
</apply>

Example

<infinity/>

Example

<apply><eq/>
  <apply><limit/>
    <bvar><ci>x</ci></bvar>
    <condition><apply><tendsto/><ci>x</ci><infinity/></apply></condition>
    <apply><divide/><cn>1</cn><ci>x</ci></apply>
  </apply>
  <cn>0</cn>
</apply>

3.13 Content Dictionary: General MathML Errors

C.1 MMLdefinition: unhandled_symbol

Description

This symbol represents the error which is raised when an application reads a symbol which is present in the mentioned content dictionary, but which it has not implemented.

When receiving such a symbol, the application should act as if it had received the MathML error object constructed from unhandled_symbol and the unhandled symbol as in the example below.

Classification

Error Specifier

Example

The application does not implement the Complex numbers:

    
<error>
    <csymbol cd="error" name="unhandled_symbol"/>
    <csymbol cd="setname1" name="C"/>
  </error>

C.2 MMLdefinition: unexpected_symbol

Description

This symbol represents the error which is raised when an application reads a symbol which is not present in the mentioned content dictionary.

When receiving such a symbol, the application should act as if it had received the MathML error object constructed from unexpected_symbol and the unexpected symbol as in the example below.

Classification

Error Specifier

Example

The application received a mistyped symbol

    
<error>
    <csymbol cd="error" name="unexpected_symbol"/>
    <csymbol cd="arith1" name="plurse"/>
  </error>

C.3 MMLdefinition: unsupported_CD

Description

This symbol represents the error which is raised when an application reads a symbol where the mentioned content dictionary is not present.

When receiving such a symbol, the application should act as if it had received the MathML error object constructed from unsupported_CD and the symbol from the unsupported Content Dictionary as in the example below.

Classification

Error Specifier

Example

The application does not know about the CD specfun

    
<error>
    <csymbol cd="error" name="unsupported_CD"/>
    <csymbol cd="specfun" name="BesselJ"/>
  </error>