Factorization of Polynomials and GCD Computations for Finding Universal Denominators

Let K be a field of characteristic zero and suppose that f: → K satisfies a recurrence of the form \[ f(n) = \sum_{i=1}^d P_i(n) f(n-i), \] for n sufficiently large, where P1(z),. . .,Pd(z) are polynomials in K[z]. Given that Pd(z) is a nonzero constant polynomial, we show that the set of n ∈ for which f(n) = 0 is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem–Mahler–Lech theorem, which assumes that f(n) satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a power series satisfying a homogeneous linear differential equation with rational function coefficients.

We give an improved algorithm for factorization, i.e. decomposition of such systems with rational function coefficients into lowerorder systems. This algorithm is based on the recent algorithm of Z. Li and F. Schwarz for construction of hyperexponential solutions. An analogue of the Loewy-Ore theory of factorization is exposed. We also prove an analogue of a result by E. Landau on uniqueness of the number and orders of irreducible factors.

Identifying integrable coupled nonlinear ordinary differential equations (ODEs) of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics. In this paper we undertake these problems and unearth two classes of integrable coupled nonlinear ODEs of arbitrary order. To achieve these goals we introduce suitable nonlocal transformations in certain linear ODEs and generate the coupled nonlinear ODEs. In particular, we show that the problem of solving these classes of coupled nonlinear ODEs of any order effectively reduces to solving a single first order nonlinear ODE. We then describe a procedure to derive explicit general solutions for the identified integrable coupled ODEs, when the above mentioned first order nonlinear ODE reduces to a Bernoulli equation. The equations which we generate and solve include the two coupled versions of modified Emden equations (in second order), coupled versions of Chazy equations (in third order), and their variants, higher dimensional coupled Ricatti and Abel’s chains, as well as a new integrable chain and higher order equations.

CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS

E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with rational functions coefficients. They were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function, and studied further by Shidlovskii in 1956. The celebrated Siegel-Shidlovskii Theorem deals with the algebraic (in)dependence of values at algebraic points of E-functions solutions of a differential system. However, somewhat paradoxically, this deep result may fail to decide whether a given E-fuction assumes an algebraic or a transcendental value at some given algebraic point. Building upon Andre's theory of E-operators, Beukers refined in 2006 the Siegel-Shidlovskii Theorem in an optimal way. In this paper, we use Beukers' work to prove the following result: there exists an algorithm which, given a transcendental E-function $f(z)$ as input, outputs the finite list of all exceptional algebraic points $\alpha$ such that $f(\alpha)$ is also algebraic, together with the corresponding list of values $f(\alpha)$. This result solves the problem of deciding whether values of E-functions at algebraic points are transcendental.

We consider a system of two hybrid vector equations containing linear difference (defined on a discrete set) and functional differential (defined on a half-axis) parts. To study it, a model system of two vector equations is chosen, one of which is linear difference with aftereffect (LDEA), and the other is a linear functional differential with aftereffect (LFDEA). Two equivalent representations of this system are shown: the first representation in the form of LFDEA, the second — in the form of LDEA. This allows us to study the stability issues of the system under consideration using the well-known results on the stability of LFDEA and LDEA. Using the results of the article [Gusarenko S. A. On the stability of a system of two linear differential equations with delayed argument // Boundary value problems. Interuniversity collection of scientific papers. Perm: PPI, 1989. P. 3–9], two examples are shown when a joint system of four equations will be stable with respect to the right side. In the first example, we use the LFDEA for which sufficient conditions for the sign-definiteness of the elements of the 2 Ч 2 Cauchy matrix function are known (in terms of the LFDEA coefficients). In the second example, LFDEA is given such that LFDEA is a system of linear ordinary differential equations (LODE) of the second order. In both cases, estimates of the components of the Cauchy matrix function are known. An exponential estimate with a negative exponent is given for the components of the Cauchy matrix function of LDEA.

The investigations presented in this paper consist of two parts; the first offers a solution, in a qualified sense, of the general linear equation in finite differences; and the second gives an analysis of the general linear differential equation with rational factors, so far as concerns its solution in series. The author observes that there does not at present exist any general method of solving linear equations in finite differences of an order higher than the first; and that with reference to such equations of the first order, we obtain insufficient forms which are intelligible only when the independent variable is an integer. It is in this qualified sense that the solutions proposed in this paper are to be taken; so that the first part of these investigations may be considered as an extension of this form of solution from the general equation of the first order to the general equation of the n th order.

Application of commutator calculus to the study of linear impulsive systems

These algorithms are probabilistic in the following sense. The time of computation depends on random choices, but the validity of the result does not depend on them. So, worst case complexity, being infinite, is meaningless and we compute average complexity. It appears that our and Cantor-Zassenhaus algorithms have the same asymptotic complexity and they are the best algorithms actually known ; with elementary multiplication and GCD computation, CantorZassenhaus algorithm is always better than ours ; with fast multiplication and GCD, it seems that ours is better, but this fact is not yet proveen.

A generalized differential transform method for linear partial differential equations of fractional order

In a recent paper, Agarwal and Pituk have considered scalar linear difference equations whose coefficients are asymptotically constant and whose corresponding perturbations are exponentially small. Using the method of generating functions and results from complex analysis, they derived an asymptotic representation for solutions, which was then applied to study the asymptotic behaviour of solutions of certain nonlinear autonomous scalar difference equations near a hyperbolic equilibrium. Here, we first show using standard matrix analysis how their results can be extended to systems of linear difference equations and that the error estimates can be made more precise. Our method also can be extended to weakly nonlinear systems. That result can in turn be used to analyze solutions of some autonomous nonlinear difference systems near an equilibrium.

This paper is concerned with the Newton—Puiseux-diagram (NP-diagram) for linear differential equations. The NP-diagram will be characterized by the NP-slopes pj and the associated NP-polynoms Pj First it is proved that the NP-slopes as well as the NP- polynoms are invariant with respect to meromorphic transformations. Using this and the fact that every system of linear differential equations can be transformed into a scalar linear differential equation of nth order, now it is possible to define the NP-diagram with NP- slopes and NP-polynoms for linear systems too. The NP-diagram is used here to get information about the growth behavior of the solutions in the neighborhood of an irregular singular point. We make use of the formal fundamental system and show that the NP-slopes and the NP-polynoms are exactly determined by the (formal) orders, the (formal) types and the (formal) maximum directions, given by the formal fundamental system, and vice versa. The connection between the NP- diagram and the (not formal) solutions of the system is made finally by the asymptotic behavior of the solutions.

Linear differential equations of higher order have useful and interesting applications, just as first-order differential equations do. We study linear differential equations of higher order in this chapter. The word linear in the chapter title should suggest that techniques for solving linear equations will be important. What is somewhat unexpected is that we have to appeal to the theory of solving polynomial equations in one variable. Though the solution technique for first-order equations gave us a complete solution in essentially one step, this is not the case here. For the first time we have to solve the homogeneous and nonhomogeneous equations separately and by different methods.

Rational Functions are used to describe the self-excited forces acting on the bridge deck in the time domain. They can be identified indirectly based on aerodynamic derivatives or directly with the free (E2RFC method) or forced vibration technique, which can significantly decrease the testing time. The approach presented herein enables the extraction of Rational Function Coefficients by testing the section model at only one wind speed. This aim is achieved by increased complexity of the forced motion compared to the previous tests, which made it possible to test a wider range of reduced velocities by adjusting the motion frequency. In this study, motion histories generated from the assumed flat spectra are used. Wind tunnel tests on a streamlined section model utilizing simultaneous vertical, horizontal and torsional vibrations were performed to extract Rational Function Coefficients associated with 3-degree-of-freedom motion. Restrictions and improvements arising from the proposed methodology are described.

The purpose of this paper is to investigate the use of rational Chebyshev (RC) collocation method for solving high-order linear ordinary differential equations with variable coefficients. Using the rational Chebyshev collocation points, this method transforms the high-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coefficients. These matrices together with the collocation method are utilized to reduce the solution of higher-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of RC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive and maintains better accuracy.