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In order for ${\mathcal C}$ to come from an algebraic group rather than a pro-algebraic one, you want ${\mathcal C}$ to be finitely generated. And for semisimplicity, you want the group to have finite ...

A nice exposition of an Erdos/Selberg-type elementary proof is given by Levinson in Amer. Math. Monthly 76 (1969) 225–245.
The proof by Daboussi as written up by Tenenbaum and Mendes-France was ...

I'm surprised that nobody has mentioned the proof using universal nets. (It can be found, e.g., in Pedersen's 'Analysis NOW' and in Bredon's 'Topology and geometry'.)
A universal net in a set X is a ...

The algebraic $K$-groups of a commutative unital ring can indeed be defined as derived functors, but one needs to work in the context of non-abelian homological algebra in the sense of A. Dold, D. ...

I have been teaching general topology for several years, but remained unsatisfied by the proofs given in the books that I based the course upon. Finally I wound up writing my own lecture notes, still ...

You might want to look at the paper "Le théorème des nombres premiers et la transformation de Fourier" by Jean-Benoît Bost, available at http://www.math.polytechnique.fr/xups/xups02-01.pdf
It gives a ...

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