In this paper we present a construction of Kari-Culik aperiodic tile set -the smallest known until now. With the help of this construction, we prove that this tileset has positive entropy. We also explain why this result was not expected. * supported by ANR project EMC NT09 555297

Cellular automata are a discrete dynamical system which models massively parallel computation. Much attention is devoted to computations with small time complexity for which the parallelism may provide further possibilities. In this paper, we investigate the ability of cellular automata related to functional computation. We introduce several functional classes of low time complexity which contain "natural" problems. We examine their inclusion relationships and emphasize that several questions arising from this functional framework are related to current ones coming from the recognition context. We also provide a negative result which explicits limits on the information transmission whose consequences go beyond the functional point of view

We investigate one dimensional partitioning of sparse matrices under a given ordering of the rows/columns. The partitioning constraint is to have load balance across processors when different parts are assigned to different processors. The load is defined as the number of rows, or columns, or the nonzeros assigned to a processor. The partitioning objective is to optimize different functions, including the well-known total communication volume arising in a distributed memory implementation of parallel sparse matrix-vector multiplication operations. The difference between our problem in this work and the general sparse matrix partitioning problem is that the parts should correspond to disjoint intervals of the given order. Whereas the partitioning problem without the interval constraint corresponds to the NP-complete hypergraph partitioning problem, the restricted problem corresponds to a polynomial-time solvable variant of the hypergraph partitioning problem. We adapt an existing dynamic programming algorithm designed for graphs to solve two related partitioning problems in graphs. We then propose graph models for a given hypergraph and a partitioning objective function so that the standard cutsize definition in the graph model exactly corresponds to the hypergraph partitioning objective function. In extensive experiments, we show that our proposed algorithm is helpful in practice. It even demonstrates performance superior to the standard hypergraph partitioners when the number of parts is high.