We describe various combinatorial aspects of the Birkhoff–Connes–Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov’s preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.

The Magnus-Derek game (also called the maximal variant of the vector game), introduced by Nedev and Muthukrishnan is the following: a token is moved around a table with n positions. In each round of the game Magnus chooses a number and then Derek chooses a direction (clockwise or counterclockwise), and the token moves that many positions into that direction. The goal of Magnus is to maximize th...

We consider the so called Magnus-Derek game, which is a two-person game played on a round table with n positions. The two players are called Magnus and Derek. Initially there is a token placed at position 0. In each round Magnus chooses a positive integer m ≤ n/2 as the distance of the targeted position from his current position for the token to move, and Derek decides a direction, clockwise or...

A classical problem in political theory is how to explain unpopular and oppressive minority rule. How can a small minority manage to dominate a much larger population against its will? How come the oppressed majority does not revolt? Two rival approaches can be distinguished: the coordination theory, according to which the power of the minority rests on coordinated action among the minority and...