31,269 research outputs found

    Periodic orbits from Δ-modulation of stable linear systems

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    The �-modulated control of a single input, discrete time, linear stable system is investigated. The modulation direction is given by cTx where c �Rn/{0} is a given, otherwise arbitrary, vector. We obtain necessary and sufficient conditions for the existence of periodic points of a finite order. Some concrete results about the existence of a certain order of periodic points are also derived. We also study the relationship between certain polyhedra and the periodicity of the �-modulated orbit

    Interlacing Log-concavity of the Boros-Moll Polynomials

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    We introduce the notion of interlacing log-concavity of a polynomial sequence {Pm(x)}m0\{P_m(x)\}_{m\geq 0}, where Pm(x)P_m(x) is a polynomial of degree m with positive coefficients ai(m)a_{i}(m). This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of Pm(x)P_m(x) interlace the ratios of consecutive coefficients of Pm+1(x)P_{m+1}(x) for any m0m\geq 0. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which implies that some classical combinatorial polynomials are interlacing log-concave.Comment: 10 page

    Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions

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    Let pˉ(n)\bar{p}(n) denote the number of overpartitions of nn. It was conjectured by Hirschhorn and Sellers that \bar{p}(40n+35)\equiv 0\ ({\rm mod\} 40) for n0n\geq 0. Employing 2-dissection formulas of quotients of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for pˉ(40n+35)\bar{p}(40n+35) modulo 5. Using the (p,k)(p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we give a proof of the congruence \bar{p}(40n+35)\equiv 0\ ({\rm mod\} 5). Combining this congruence and the congruence \bar{p}(4n+3)\equiv 0\ ({\rm mod\} 8) obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we give a proof of the conjecture of Hirschhorn and Sellers.Comment: 11 page