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- In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function. The original examples of distributions occur, unnamed, as functions φ on Q/Z satisfying Such distributions are called ordinary distributions. They also occur in p-adic integration theory in Iwasawa theory. Let ... → Xn+1 → Xn → ... be a projective system of finite sets with surjections, indexed by the natural numbers, and let X be their projective limit. We give each Xn the discrete topology, so that X is compact. Let φ = (φn) be a family of functions on Xn taking values in an abelian group V and compatible with the projective system: for some weight function w. The family φ is then a distribution on the projective system X. A function f on X is "locally constant", or a "step function" if it factors through some Xn. We can define an integral of a step function against φ as The definition extends to more general projective systems, such as those indexed by the positive integers ordered by divisibility. As an important special case consider the projective system Z/nZ indexed by positive integers ordered by divisibility. We identify this with the system (1/n)Z/Z with limit Q/Z. For x in R we let ⟨x⟩ denote the fractional part of x normalised to 0 ≤ ⟨x⟩ < 1, and let {x} denote the fractional part normalised to 0 < {x} ≤ 1. (en)
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- 7465 (xsd:nonNegativeInteger)
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- In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function. The original examples of distributions occur, unnamed, as functions φ on Q/Z satisfying Such distributions are called ordinary distributions. They also occur in p-adic integration theory in Iwasawa theory. for some weight function w. The family φ is then a distribution on the projective system X. (en)
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- Distribution (number theory) (en)
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