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## 9.37 Kernel

A kernel is a symbolic function application (such as sin(x+ y)) or a symbol (such as x). More precisely, a non-symbol kernel over a set S is an operator applied to a given list of arguments from S. The operator has type BasicOperator (see BasicOperatorXmpPage ) and the kernel object is usually part of an expression object (see ExpressionXmpPage ).

Kernels are created implicitly for you when you create expressions.

x :: Expression Integer

x

Type: Expression Integer

You can directly create a ``symbol'' kernel by using the kernel kernel Kernel operation.

kernel x

x

Type: Kernel Expression Integer

This expression has two different kernels.

sin(x) + cos(x)

sin(x)+cos(x)

Type: Expression Integer

The operator kernels kernels Expression returns a list of the kernels in an object of type Expression.

kernels %

[sin(x),cos(x)]

Type: List Kernel Expression Integer

This expression also has two different kernels.

sin(x)**2 + sin(x) + cos(x)

sin(x)2+sin(x)+cos(x)

Type: Expression Integer

The sin(x) kernel is used twice.

kernels %

[sin(x),cos(x)]

Type: List Kernel Expression Integer

An expression need not contain any kernels.

kernels(1 :: Expression Integer)

[]

Type: List Kernel Expression Integer

If one or more kernels are present, one of them is designated the main kernel.

mainKernel(cos(x) + tan(x))

tan(x)

Type: Union(Kernel Expression Integer,...)

Kernels can be nested. Use height height Kernel to determine the nesting depth.

height kernel x

1

Type: PositiveInteger

This has height 2 because the x has height 1 and then we apply an operator to that.

height mainKernel(sin x)

2

Type: PositiveInteger
height mainKernel(sin cos x)

3

Type: PositiveInteger
height mainKernel(sin cos (tan x + sin x))

4

Type: PositiveInteger

Use the operator operator Kernel operation to extract the operator component of the kernel. The operator has type BasicOperator.

operator mainKernel(sin cos (tan x + sin x))

sin

Type: BasicOperator

Use the name name Kernel operation to extract the name of the operator component of the kernel. The name has type Symbol. This is really just a shortcut for a two-step process of extracting the operator and then calling name name BasicOperator on the operator.

name mainKernel(sin cos (tan x + sin x))

sin

Type: Symbol

Axiom knows about functions such as sin, cos and so on and can make kernels and then expressions using them. To create a kernel and expression using an arbitrary operator, use operator operator BasicOperator.

Now f can be used to create symbolic function applications.

f := operator 'f

f

Type: BasicOperator
e := f(x, y, 10)

f(x,y,10)

Type: Expression Integer

Use the is?is?Kernel operation to learn if the operator component of a kernel is equal to a given operator.

is?(e, f)

true

Type: Boolean

You can also use a symbol or a string as the second argument to is?is?Kernel.

is?(e, 'f)

true

Type: Boolean

Use the argument argument Kernel operation to get a list containing the argument component of a kernel.

argument mainKernel e

[x,y,10]

Type: List Expression Integer

Conceptually, an object of type Expression can be thought of a quotient of multivariate polynomials, where the ``variables'' are kernels. The arguments of the kernels are again expressions and so the structure recurses. See ExpressionXmpPage for examples of using kernels to take apart expression objects.

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