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Carlos E Arreche - One of the best experts on this subject based on the ideXlab platform.
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computation of the Difference differential galois group and differential relations among solutions for a second order Linear Difference Equation
Communications in Contemporary Mathematics, 2017Co-Authors: Carlos E ArrecheAbstract:We apply the Difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous Linear Difference Equation y(x + 2) + a(x)y(x + 1) + b(x)y(x) = 0, where the coefficients a(x),b(x) ∈ℚ(x) are rational functions in x with coefficients in ℚ. We develop algorithms to compute the Difference-differential Galois group associated to such an Equation, and show how to deduce the differential-algebraic relations among the solutions from the defining Equations of the Galois group.
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computation of the Difference differential galois group and differential relations among solutions for a second order Linear Difference Equation
arXiv: Commutative Algebra, 2016Co-Authors: Carlos E ArrecheAbstract:We apply the Difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous Linear Difference Equation of the form $ y(x+2)+a(x)y(x+1)+b(x)y(x)=0,$ where the coefficients $a(x),b(x)\in \bar{\mathbb{Q}}(x)$ are rational functions in $x$ with coefficients in $\bar{\mathbb{Q}}$. We develop algorithms to compute the Difference-differential Galois group associated to such an Equation, and show how to deduce the differential-algebraic relations among the solutions from the defining Equations of the Galois group.
S A Abramov - One of the best experts on this subject based on the ideXlab platform.
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factorization of polynomials and gcd computations for finding universal denominators
Computer Algebra in Scientific Computing, 2010Co-Authors: S A Abramov, A Gheffar, D E KhmelnovAbstract:We discuss the algorithms which, given a Linear Difference Equation with rational function coefficients over a field k of characteristic 0, compute a polynomial U(x) ∈ k[x] (a universal denominator) such that the denominator of each of rational solutions (if exist) of the given Equation divides U(x). We consider two types of such algorithms. One of them is based on constructing a set of irreducible polynomials that are candidates for divisors of denominators of rational solutions, and on finding a bound for the exponent of each of these candidates (the full factorization of polynomials is used). The second one is related to earlier algorithms for finding universal denominators, where the computation of gcd was used instead of the full factorization. The algorithms are applicable to scalar Equations of arbitrary orders as well as to systems of first-order Equations. A complexity analysis and a time comparison of the algorithms implemented in Maple are presented.
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rational solutions of Linear Difference and q Difference Equations with polynomial coefficients
International Symposium on Symbolic and Algebraic Computation, 1995Co-Authors: S A AbramovAbstract:A simple algorithm is suggested for the construction of a polynomial divisible by the denominator of any rational solution of the Linear Difference Equation.
Elena M. Parilina - One of the best experts on this subject based on the ideXlab platform.
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Soil acidity adaptive control problem
Stochastic Environmental Research and Risk Assessment, 2015Co-Authors: Victor P. Yakushev, Vladimir V. Karelin, Vladimir M. Bure, Elena M. ParilinaAbstract:Problem of soil acidity regularization is modeled as stochastic adaptive control problem with a Linear Difference Equation of the dynamics of a field pH level. Stochastic component in the Equation represents an individual time variability of soil acidity of an elementary section. We use Bayesian approach to determine a posteriori probability density function of the unknown parameters of the stochastic transition process. The Kullback–Leibler information divergence is used as a measure of Difference between true distribution and its estimation. Algorithm for the construction of an adaptive stabilizing control in such a Linear control system is proposed in the paper. Numerical realization of the algorithm is represented for a problem of a field soil acidity control.
I P Stavroulakis - One of the best experts on this subject based on the ideXlab platform.
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optimal oscillation criteria for first order Difference Equations with delay argument
Pacific Journal of Mathematics, 2008Co-Authors: George E Chatzarakis, Roman Koplatadze, I P StavroulakisAbstract:Consider the first order Linear Difference Equation 1u(k)+ p(k)u(( k)) = 0, k 2 N, where 1u(k) = u(k+1) u(k), p : N ! R+, : N ! N, ( k) k 2 and limk!+1 ( k) = +1. Optimal conditions for the oscillation of all proper solutions of this Equation are established. The results lead to a sharp oscillation condition, when k ( k) ! +1 as k ! +1. Examples illustrating the results are given.
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oscillation criteria of first order Linear Difference Equations with delay argument
Nonlinear Analysis-theory Methods & Applications, 2008Co-Authors: George E Chatzarakis, Roman Koplatadze, I P StavroulakisAbstract:Abstract This paper presents new sufficient conditions for the oscillation of all proper solutions of the first order Linear Difference Equation with delay argument Δ u ( k ) + p ( k ) u ( τ ( k ) ) = 0 , k ∈ N , where Δ u ( k ) = u ( k + 1 ) − u ( k ) , p : N → R + , τ : N → N and lim k → + ∞ τ ( k ) = + ∞ . Examples illustrating the results are given. It is to be pointed out that this is the first paper dealing with the oscillatory behaviour of the Equation in the case of a general delay argument τ ( k ) .
Victor P. Yakushev - One of the best experts on this subject based on the ideXlab platform.
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Soil acidity adaptive control problem
Stochastic Environmental Research and Risk Assessment, 2015Co-Authors: Victor P. Yakushev, Vladimir V. Karelin, Vladimir M. Bure, Elena M. ParilinaAbstract:Problem of soil acidity regularization is modeled as stochastic adaptive control problem with a Linear Difference Equation of the dynamics of a field pH level. Stochastic component in the Equation represents an individual time variability of soil acidity of an elementary section. We use Bayesian approach to determine a posteriori probability density function of the unknown parameters of the stochastic transition process. The Kullback–Leibler information divergence is used as a measure of Difference between true distribution and its estimation. Algorithm for the construction of an adaptive stabilizing control in such a Linear control system is proposed in the paper. Numerical realization of the algorithm is represented for a problem of a field soil acidity control.