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8DIO – Dubstep (Kontakt) Download. 8Dio.Dubstep (Kontakt) download online full. Nomad Factory Integral Studio Pack 3 V5.1.0 R3 (Win-Osx) download.Sobolev p-adic function

In mathematics, specifically p-adic number theory, a Sobolev p-adic function, or a smooth p-adic function (after Laurent Schwartz) is a certain kind of real analytic p-adic function.

Definition
An integer-valued function on the principal series of the p-adic integers is a p-adic function of Sobolev class 0 if there exists a nonnegative integer for which

where denotes the supremum norm over the ball of radius. Note that this only requires to be nonnegative, whereas the zero function would have to be nonnegative for every.

Example
The usual example of a function of Sobolev class 0 is the characteristic function of the ball of radius centered at, i.e.

where denotes the principal series of the p-adic integers. This function is easy to see to be of Sobolev class 0.

Normalization
Because depends not only on but on, the notion of Sobolev class must be normalized. Namely, if, then

is a function of Sobolev class 1. One cannot take higher Sobolev classes, as is evident from the lack of continuity of at, though one might wish to restrict oneself to a certain Sobolev class if one is only concerned with the p-adic properties of the function.

Subtraction
Note that for any two functions and with, the difference is of Sobolev class 0 if and only if both and are. For this reason, the function defined by

with the convention that if, is also of
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