31,269 research outputs found

### Periodic orbits from Î”-modulation of stable linear systems

The ÃŽï¿½-modulated control of a single input, discrete time, linear stable system is investigated. The modulation direction is given by cTx where c Ã¢ï¿½ï¿½Rn/{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 ÃŽï¿½-modulated orbit

### Interlacing Log-concavity of the Boros-Moll Polynomials

We introduce the notion of interlacing log-concavity of a polynomial sequence
$\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with
positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be
interlacing log-concave if the ratios of consecutive coefficients of $P_m(x)$
interlace the ratios of consecutive coefficients of $P_{m+1}(x)$ for any $m\geq
0$. 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

Let $\bar{p}(n)$ denote the number of overpartitions of $n$. It was
conjectured by Hirschhorn and Sellers that \bar{p}(40n+35)\equiv 0\ ({\rm
mod\} 40) for $n\geq 0$. Employing 2-dissection formulas of quotients of theta
functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating
function for $\bar{p}(40n+35)$ modulo 5. Using the $(p, k)$-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

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