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In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
== Definition ==
If
f
,
g
:
N
→
C
{\displaystyle f,g:\mathbb {N} \to \mathbb {C} }
are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution f ∗ g is a new arithmetic function defined by:
(
f
∗...
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Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural to define the entropy of such function $f$ as follows:
$$H(f) = -\sum _{x \in \mathbb{Z}_2^n} f(x...
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In category theory there is a 'folk' model structure on the category Cat, where the weak equivalences are the equivalences of categories. There is a similar model structure on 2Cat, with weak equivalences being equivalences of 2-categories (weak ones, I presume) The former was not written down fo...
