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Calling Sequence
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Power(a, b, ...)
a ^ b
a &^ b
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Parameters
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a, b, ...
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ordinals, nonnegative integers, or polynomials with positive integer coefficients
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Description
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The Power and ^ calling sequences perform exponentiation of the given ordinal numbers according to the rules of ordinal arithmetic. Let , where is a positive integer and or in the strict ordering of ordinals.
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If is a positive integer, then .
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If is a positive integer and , where is a nonnegative integer, then .
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If and , then .
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If , then .
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If more than two arguments are specified, the powering will be performed right-associatively, that is, .
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Mathematically, exponentiation of two ordinals corresponds to the set of all functions , between the corresponding well-orderings and , with finite support, such that for only finitely many , together with the ordering defined by:
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In the two-argument case, if are both nonzero, and at least one of them is an ordinal data structure, that is, an ordinal number greater or equal to , then the result is an ordinal data structure. Otherwise, the result is a nonnegative integer or a polynomial with positive integer coefficients.
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The &^ calling sequence is the inert form of ordinal exponentiation. No actual exponentiation is performed, but the result will be rendered as an inert power, with parentheses around the first argument.
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Applying the value command will turn the inactive &. operator into the active . operator, causing the ordinal multiplication to be computed as described above.
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The first argument can be a parametric ordinal. If it cannot be determined whether its leading or trailing coefficient is nonzero, an error may be raised.
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Examples
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| (1) |
>
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>
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>
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>
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| (9) |
The inert exponentiation operator is useful for display purposes:
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Parametric examples:
>
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| (11) |
| (12) |
>
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| (13) |
| (15) |
| (16) |
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Compatibility
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The Ordinals[Power], Ordinals[`^`] and &^ commands were introduced in Maple 2015.
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