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    Fixed Parameter Undecidability for Wang Tilesets

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    Deciding if a given set of Wang tiles admits a tiling of the plane is decidable if the number of Wang tiles (or the number of colors) is bounded, for a trivial reason, as there are only finitely many such tilesets. We prove however that the tiling problem remains undecidable if the difference between the number of tiles and the number of colors is bounded by 43. One of the main new tool is the concept of Wang bars, which are equivalently inflated Wang tiles or thin polyominoes.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

    On the qq-log-convexity conjecture of Sun

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    In his study of Ramanujan-Sato type series for 1/π1/\pi, Sun introduced a sequence of polynomials Sn(q)S_n(q) as given by Sn(q)=k=0n(nk)(2kk)(2(nk)nk)qk,S_n(q)=\sum\limits_{k=0}^n{n\choose k}{2k\choose k}{2(n-k)\choose n-k}q^k, and he conjectured that the polynomials Sn(q)S_n(q) are qq-log-convex. By imitating a result of Liu and Wang on generating new qq-log-convex sequences of polynomials from old ones, we obtain a sufficient condition for determining the qq-log-convexity of self-reciprocal polynomials. Based on this criterion, we then give an affirmative answer to Sun's conjecture

    An inequality of W. L. Wang and P. F. Wang

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    In this note we present a proof of the inequality Hn/H′n≤Gn/G′n where Hn and Gn (resp. H′n and G′n) denote the weighted harmonic and geometric means of x1,…,xn (resp. 1−x1,…,1−xn) with xi∈(0,1/2], i=1,…,n