Journal of Difference Equations and Applications


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Publications in this journal

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    ABSTRACT: Recently, a new class of words, denoted by [Inline formula], was shown to be in bijection with a subset of the Dyck paths of length [Inline formula] having cardinality the Catalan number [Inline formula]. Here, we consider statistics on [Inline formula] recording the number of occurrences of a letter [Inline formula]. In the cases [Inline formula] and [Inline formula], we are able to determine explicit expressions for the number of members of [Inline formula] containing a given number of zeros or ones, which generalizes the prior result. To do so, we make use of recurrences to derive a functional equation satisfied by the generating function, which we solve by a new method employing Chebyshev polynomials. In the case [Inline formula], our result is equivalent to a prior one concerning the distribution of the initial rise statistic on Dyck paths. Recurrences and generating function formulas are also provided in the case of general [Inline formula].
    Journal of Difference Equations and Applications 09/2014; 20(11):1568--1582.
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    ABSTRACT: Infinite Leslie matrices, introduced by Demetrius 40 years ago, are mathematical models of age-structured populations defined by a countable infinite number of age classes. This article is concerned with determining solutions of the discrete dynamical system in finite time. We address this problem by appealing to the concept of kneading matrices and kneading determinants. Our analysis is applicable not only to populations models, but also to models of self-reproducing machines and self-reproducing computer programs. The dynamics of these systems can also be described in terms of infinite Leslie matrices.
    Journal of Difference Equations and Applications 05/2014;
  • Journal of Difference Equations and Applications 12/2012; 18(12):1951-1966.
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    ABSTRACT: Let X be a complex Banach space and q ≥ 2 be a fixed integer number. Let be a q-periodic discrete evolution family generated by the (X)-valued, q-periodic sequence (A n ). We prove that the solution of the following discrete problem is bounded (uniformly with respect to the parameter μ ) for each vector b X if and only if the Poincare map U(q,0) is stable.
    Journal of Difference Equations and Applications 09/2012; 18(9):1435-1441.
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    ABSTRACT: A modified numerical method was used by authors for solving 1D Stefan problem. In this paper a modified method is proposed with difference formulae and different methods of calculating the variable time step, which are deduced from Taylor series expansions of different conditions at the boundary. Also an extrapolation formula for the solution at the first point at the right of the computational domain is proposed. The numerical results are compared with those obtained from other methods.
    Journal of Difference Equations and Applications 09/2012; 18(09):1443-1455.
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    ABSTRACT: Exact finite difference schemes and non-standard finite difference schemes are constructed for the first-order differential equation , for and . In particular, we show that the central finite difference scheme is an exact scheme for the differential equation .
    Journal of Difference Equations and Applications 09/2012; 18(9):1511-1517.
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    ABSTRACT: Stochastic difference equations and a stochastic partial differential equation (SPDE) are simultaneously derived for the time-dependent neutron angular density in a general three-dimensional medium where the neutron angular density is a function of position, direction, energy and time. Special cases of the equations are given, such as transport in one-dimensional plane geometry with isotropic scattering and transport in a homogeneous medium. The stochastic equations are derived from basic principles, i.e. from the changes that occur in a small time interval. Stochastic difference equations of the neutron angular density are constructed, taking into account the inherent randomness in scatters, absorptions and source neutrons. As the time interval decreases, the stochastic difference equations lead to a system of Itô stochastic differential equations. As the energy, direction, and position intervals decrease, an SPDE is derived for the neutron angular density. Comparisons between numerical solutions of the stochastic difference equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.
    Journal of Difference Equations and Applications 08/2012; 18(8):1267-1285.
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    ABSTRACT: This paper presents a proof of almost sure asymptotic stability of trivial solution of stochastic systems of linear difference equations driven by -integrable martingale-type noise. For this purpose, we exploit a convergence theorem for non-negative semi-martingale decompositions and verify a practical criteria based on non-random eigenvalues. Two 2D examples illustrate the applicability of obtained criteria.
    Journal of Difference Equations and Applications 08/2012; 18(8):1333-1343.
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    ABSTRACT: It is known that if we apply Newton's method to the complex function , with , then the immediate basin of attraction of the roots of P has finite area. In this paper, we show that under certain conditions on the polynomial P, if , then there is at least one immediate basin of attraction having infinite area.
    Journal of Difference Equations and Applications 06/2012; 18(6):1067-1076.
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    ABSTRACT: Let G be a finite connected graph. Suppose is a map homotopic to the identity that permutes the vertices. For such a map, a rotation matrix is defined and the basic properties of this matrix are given. It is shown that this matrix generalizes some of the information given by the rotation interval, which is defined when the graph is a circle, to more general graphs.
    Journal of Difference Equations and Applications 06/2012; 18(6):1033-1041.
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    ABSTRACT: We study the existence of positive solutions with respect to a cone for a nonlinear system with second-order differences, subject to some -point boundary conditions.
    Journal of Difference Equations and Applications 05/2012; 18(5):865-877.
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    ABSTRACT: The second-order non-autonomous difference equation where h>0 is a parameter and f is continuous and has three real zeros L 0 < 0 < L is investigated. The equation is a discretization of differential equations arising in hydrodynamics or in the nonlinear field theory. This paper provides conditions for f which guarantee that the equation can have four types of solutions – escape, homoclinic, damped and non-monotonous.
    Journal of Difference Equations and Applications 05/2012; 18(5):895-907.
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    ABSTRACT: We describe the sequences given by the non-autonomous second-order Lyness difference equations , where is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditions are also positive. We also show an interesting phenomenon of the discrete dynamical systems associated with some of these difference equations: the existence of one oscillation of their associated rotation number functions. This behaviour does not appear for the autonomous Lyness difference equations.
    Journal of Difference Equations and Applications 05/2012; 18(5):849-864.