Periodicity and arithmetic-periodicity in hexadecimal games

Periodicity and Almost-periodicity

Ultime periodicity in chip-firing games

One can study variations of this discrete dynamical system : applying the rule sequentially (on exactly one vertex at each time step), the restriction to undirected graphs or other families of graphs, the extension to multi-graphs, etc. Such models are called chip-firing games (CFG). Some original definitions where given in [19] and [1], but the two references to this respect are [4] for the un...

On the periodicity of graph games

Starting with the empty graph on p vertices, two players alternately add edges until the graph has some property P. The last player to move loses the P-avoidance game (there is also a P-achievement game, where the last player to move wins). If P is a property enjoyed by the complete graph on p vertices then one player must win after a finite number of moves. Assuming both players play optimally...

Poincaré Duality and Periodicity, Ii. James Periodicity

Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension ΣK so that the resulting space satisfies Poincaré duality. When this is possible, we say that ΣK is a spine. We introduce the notion of quadratic self duality and show that if K is quadratically self dual, then ΣK is a spine whenever j is a suitable power of two. The ...

Arithmetic Progressions in Sumsets and L-almost-periodicity

We prove results about the L-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L, and gives a very short proof of a theorem of Green that if A and B are subsets of {1, . . . , N} of sizes αN and βN then A + B contains an arithmetic progression of length at least exp ( c(αβ logN) − log logN ) . Another almost...