René Ndoundam

No polynomial bound for the period of the parallel chip firing game on graphs

Abstract The following (solitaire) game is considered: Initially each node of a simple, connected, finite graph contains a finite number of chips. A move consists in firing all nodes with at least as many chips as their degree, where firing a node corresponds to sending one of the nodes chips to each one of the nodes neighbors.

Exponential transient length generated by a neuronal recurrence equation

We study the sequences generated by neuronal recurrence equations of the form x(n)=1?j=1kajx(n?j)??], where k is the size of memory (k represents the number of previous states x(n?1),x(n?2),?,x(n?k) which intervene in the calculation of x(n)). We are interested in the number of steps (transient length) from an initial configuration to the cycle, where the length of the cycle represents the period. We show that under certain hypotheses it is possible to build a neuronal recurrence equation of memory size (s+1)6m, whose dynamics contains an evolution of transient length (s+1)(3m+1+lcm(p0,p1,?,ps?1,3m?1)) and a cycle of length (s+1)lcm(p0,p1,?,ps?1), where lcm denotes the least common multiple and p0,p1,?,ps?1 are prime numbers lying between 2m and 3m.

Routing automorphisms of the hypercube

We present an online algorithm for routing the automorphisms (BPC permutations) of the queueless MIMD hypercube. The routing algorithm has the virtue of being executed by each node of the hypercube without knowing the state of the others nodes. The algorithm is also vertex and link-contention free. We show, using the proposed algorithm, that BPC permutations are arbitrarily routable in the considered communication model.

Parallel Chip Firing Game Associated with n-cube Edges Orientations

We study the cycles generated by the chip firing game associated with n-cube orientations. We consider a particular class of partitions of vertices of n-cubes called left cyclic partitions that induce parallel periodic evolutions. Using this combinatorical model, we show that cycles generated by parallel evolutions are of even lengths from 2 to 2 n on H n (n≥ 1), and of odd lengths different from 3 and ranging from 1 to 2 n − 1-1 on H n (n≥ 4). However, the question weather there exist parallel evolutions with period greater that 2 n remains opened.