195 research outputs found

    Disjointness properties for Cartesian products of weakly mixing systems

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    For n≥1n\geq 1 we consider the class JP(nn) of dynamical systems whose every ergodic joining with a Cartesian product of kk weakly mixing automorphisms (k≥nk\geq n) can be represented as the independent extension of a joining of the system with only nn coordinate factors. For n≥2n\geq 2 we show that, whenever the maximal spectral type of a weakly mixing automorphism TT is singular with respect to the convolution of any nn continuous measures, i.e. TT has the so-called convolution singularity property of order nn, then TT belongs to JP(n−1n-1). To provide examples of such automorphisms, we exploit spectral simplicity on symmetric Fock spaces. This also allows us to show that for any n≥2n\geq 2 the class JP(nn) is essentially larger than JP(n−1n-1). Moreover, we show that all members of JP(nn) are disjoint from ergodic automorphisms generated by infinitely divisible stationary processes.Comment: 24 pages, corrected versio

    Locally identifying coloring in bounded expansion classes of graphs

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    A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper bound for the lid-chromatic number of planar graphs. This answers in a positive way a question of Esperet et al [L. Esperet, S. Gravier, M. Montassier, P. Ochem and A. Parreau. Locally identifying coloring of graphs. Electronic Journal of Combinatorics, 19(2), 2012.]

    Joining primeness and disjointness from infinitely divisible systems

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    We show that ergodic dynamical systems generated by infinitely divisible stationary processes are disjoint in the sense of Furstenberg with distally simple systems and systems whose maximal spectral type is singular with respect to the convolution of any two continuous measures.Comment: 15 page

    A new approach to the 22-regularity of the â„“\ell-abelian complexity of 22-automatic sequences

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    We prove that a sequence satisfying a certain symmetry property is 22-regular in the sense of Allouche and Shallit, i.e., the Z\mathbb{Z}-module generated by its 22-kernel is finitely generated. We apply this theorem to develop a general approach for studying the â„“\ell-abelian complexity of 22-automatic sequences. In particular, we prove that the period-doubling word and the Thue--Morse word have 22-abelian complexity sequences that are 22-regular. Along the way, we also prove that the 22-block codings of these two words have 11-abelian complexity sequences that are 22-regular.Comment: 44 pages, 2 figures; publication versio

    The switch operators and push-the-button games: a sequential compound over rulesets

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    We study operators that combine combinatorial games. This field was initiated by Sprague-Grundy (1930s), Milnor (1950s) and Berlekamp-Conway-Guy (1970-80s) via the now classical disjunctive sum operator on (abstract) games. The new class consists in operators for rulesets, dubbed the switch-operators. The ordered pair of rulesets (R 1 , R 2) is compatible if, given any position in R 1 , there is a description of how to move in R 2. Given compatible (R 1 , R 2), we build the push-the-button game R 1 R 2 , where players start by playing according to the rules R 1 , but at some point during play, one of the players must switch the rules to R 2 , by pushing the button ". Thus, the game ends according to the terminal condition of ruleset R 2. We study the pairwise combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we prove that standard periodicity results for Subtraction games transfer to this setting, and we give partial results for a variation of Domineering, where R 1 is the game where the players put the domino tiles horizontally and R 2 the game where they play vertically (thus generalizing the octal game 0.07).Comment: Journal of Theoretical Computer Science (TCS), Elsevier, A Para{\^i}tr

    On powers of interval graphs and their orders

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    It was proved by Raychaudhuri in 1987 that if a graph power Gk−1G^{k-1} is an interval graph, then so is the next power GkG^k. This result was extended to mm-trapezoid graphs by Flotow in 1995. We extend the statement for interval graphs by showing that any interval representation of Gk−1G^{k-1} can be extended to an interval representation of GkG^k that induces the same left endpoint and right endpoint orders. The same holds for unit interval graphs. We also show that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the main result of this note, follows from earlier results of [G. Agnarsson, P. Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003]. This version is updated accordingl
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