Section9.3

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## 9.3 BasicOperator

A basic operator is an object that can be symbolically applied to a list of arguments from a set, the result being a kernel over that set or an expression. In addition to this section, please see ExpressionXmpPage and KernelXmpPage for additional information and examples.

You create an object of type BasicOperator by using the operatoroperatorBasicOperator operation. This first form of this operation has one argument and it must be a symbol. The symbol should be quoted in case the name has been used as an identifier to which a value has been assigned.

A frequent application of BasicOperator is the creation of an operator to represent the unknown function when solving a differential equation.

Let y be the unknown function in terms of x.

y := operator 'y

y

Type: BasicOperator

This is how you enter the equation y'' + y' + y = 0.

deq := D(y x, x, 2) + D(y x, x) + y x = 0

y′′(x)+y′(x)+y(x)=0

Type: Equation Expression Integer

To solve the above equation, enter this.

solve(deq, y, x)

[particular=0, basis=[cos(x 3 2)e(-x 2),e(-x 2)sin(x 3 2)]]

Type: Union(Record(particular: Expression Integer, basis: List Expression Integer),...)

Use the single argument form of operatoroperatorBasicOperator (as above) when you intend to use the operator to create functional expressions with an arbitrary number of arguments

Nary means an arbitrary number of arguments can be used in the functional expressions.

nary? y

true

Type: Boolean
unary? y

false

Type: Boolean

Use the two-argument form when you want to restrict the number of arguments in the functional expressions created with the operator.

This operator can only be used to create functional expressions with one argument.

opOne := operator('opOne, 1)

opOne

Type: BasicOperator
nary? opOne

false

Type: Boolean
unary? opOne

true

Type: Boolean

Use arityarityBasicOperator to learn the number of arguments that can be used. It returns "false" if the operator is nary.

arity opOne

1

Type: Union(NonNegativeInteger,...)

Use namenameBasicOperator to learn the name of an operator.

name opOne

opOne

Type: Symbol

Use is? is? BasicOperator to learn if an operator has a particular name.

is?(opOne, 'z2)

false

Type: Boolean

You can also use a string as the name to be tested against.

is?(opOne, "opOne")

true

Type: Boolean

You can attached named properties to an operator. These are rarely used at the top-level of the Axiom interactive environment but are used with Axiom library source code.

By default, an operator has no properties.

properties y

table()

Type: AssociationList(String,None)

The interface for setting and getting properties is somewhat awkward because the property values are stored as values of type None.

Attach a property by using setPropertysetPropertyBasicOperator.

setProperty(y, "use", "unknown function" :: None )

y

Type: BasicOperator
properties y

table("use"=NONE)

Type: AssociationList(String,None)

We know the property value has type String.

property(y, "use") :: None pretend String

"unknownfunction"

Type: String

Use deleteProperty! deleteProperty! BasicOperator to destructively remove a property.

deleteProperty!(y, "use")

y

Type: BasicOperator
properties y

table()

Type: AssociationList(String,None)
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