具有正负系数的中立型时滞微分方程的零点分布

考虑具有正负系数的中立型时滞微分方程 [x(t) C(t)x(t-γ))' P(t)x(t-(?))-Q(t)x(t-σ)=0,其中 .本文给出了上述方程的零点距估计和方程所有解都振动的充分条件,并对一些已有的零点距的估计结果作了改进.     

ON A PROBLEM IN COMPLEX OSCILLATION THEORY OF PERIODIC HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

In this article, the zeros of solutions of differential equation f(k)(z)+A(z)f(z) = 0, (*) are studied, where k 2, A(z) = B(ez), B(ζ) = g1(1/ζ) + g2(ζ), g1 and g2 being entire functions with g2 transcendental and σ(g2) not equal to a positive integer or infinity. It is shown that any linearly independent solutions f1, f2, . . . , fk of Eq.(*) satisfy λe(f1 . . . fk) ≥σ(g2) under the condition that fj(z) and fj(z+ 2πi) (j = 1, . . . , k) are linearly dependent.     

具有正负系数的连续变量的差分方程解的零点分布

考虑具有正负系数的连续变量的差分方程 x(t)-x(t-γ)+P(t)x(t-τ)-Q(t)x(t-σ)=0,其中P,Q∈C([t_0,∞),R~+),τ,σ,γ∈(0,∞)。本文给出了上述方程解的零点分布及方程所有解振动的充分条件并改进和推广了已有的结果。     

Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

The characteristic equation of a system of delay differential equations (DDEs) is a nonlinear equation with infinitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e., stability determining, zeros of the characteristic equation. The algorithm is based on the application of subspace iteration on the time integration operator of the system or its variational equations. The computed zeros provide insight into the system’s behaviour, can be used for robust bifurcation detection and for efficient indirect calculation of bifurcation points. This revised version was published online in June 2006 with corrections to the Cover Date.     

Characteristic frequencies of zeros of a sum of two harmonic oscillations

We completely describe the characteristic frequencies of zeros of a sum of two harmonic oscillations with distinct frequencies and partly describe the characteristic frequencies of zeros of sums of arbitrarily many oscillations. As a corollary, we show that the set of characteristic frequencies of zeros of a linear equation can be infinite and moreover have continuum measure even in the case of an equation with constant coefficients.     

Solving singular convolution equations using the inverse Fast Fourier Transform

The inverse Fast Fourier Transform is a common procedure to solve a convolution equation provided the transfer function has no zeros on the unit circle. In our paper we generalize this method to the case of a singular convolution equation and prove that if the transfer function is a trigonometric polynomial with simple zeros on the unit circle, then this method can be extended.     

Zeros of Racah coefficients and the pell equation

The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation. The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.The results of this paper were presented in an invited talk by one of us (JDL) at the NSF-CBMS Regional Conference on Special Functions, Physics and Computer Algebra, May 20–24, 1985, Arizona State University, Tempe, AZ.     

Integral dispersion equation method to solve a nonlinear boundary eigenvalue problem

In this work a nonlinear eigenvalue problem for a nonlinear autonomous ordinary differential equation of the second order is considered. This problem describes the process of propagation of transverse-electric electromagnetic waves along a plane dielectric waveguide with nonlinear permittivity. We demonstrate, as far as we know, a new method that allows one to derive an equation w.r.t. spectral parameter (the dispersion equation) which contains all necessary information about the eigenvalues. The method is based on a simple idea that the distance between zeros of a periodic solution to the differential equation is the same for the adjacent zeros. This method has no connections with the perturbation theory or the notion of a bifurcation point. Theorem of equivalence between the eigenvalue problem and the dispersion equation is proved. Periodicity of the eigenfunctions is proved, a formula for the period is found, and zeros of the eigenfunctions are determined. The formula for the distance between adjacent zeros of any eigenfunction is given. Also theorems of existence and localization of the eigenvalues are proved.     

VALUE DISTRIBUTION OF MEROMORPHIC SOLUTIONS OF CERTAIN NON-LINEAR DIFFERENCE EQUATIONS

In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients. For the above equation, the order of growth, the exponents of convergence of zeros and poles of its transcendental meromorphic solution f(z), and the exponents of convergence of poles of difference ?f(z) and divided difference?f(z)/f(z)are estimated. Furthermore, we study the forms of rational solutions of the above equation.     

Lagrangian flows and the one-dimensional Peano phenomenon for ODEs

We consider the one-dimensional ordinary differential equation with a vector field which is merely continuous and nonnegative, and satisfies a condition on the amount of zeros. Although it is classically known that this problem lacks uniqueness of classical trajectories, we show that there is uniqueness for the so-called regular Lagrangian flow (by now usual notion of flow in nonsmooth situations), as well as uniqueness of distributional solutions for the associated continuity equation. The proof relies on a space reparametrization argument around the zeros of the vector field.     

高阶齐次线性微分方程解的零点

研究了一类高阶齐次线性微分方程解的零点收敛指数,并得到当方程的系数A_0为整函数,其泰勒展式为缺项级数,并且A_0起控制作用时,方程f~((k))+A_(k-2)f~((k-2))+…+A_1f′+A_0f=0的任意两个线性无关解f_1,f_2满足max{λ(f_1),λ(f_2)}=∞,其中λ(f)表示亚纯函数.f的零点收敛指数.     

Nullstellen von Real- und Imaginärteil der Lösungen linearer Differentialgleichungen 2. Ordnung

This paper is concerned with complex Sturmtype comparison theorems. Especially it is shown that the location of the zeros of a nontrivial solution of a second order linear differential equation with complex-valued coefficients gives information about the distribution of the zeros of the real and imaginary part of any solution of certain other complex second order linear differential equations.     

二阶周期线性微分方程解的几个性质

In this paper,the zeros of solutions of periodic second order linear differential equation y ＋ Ay = 0,where A（z） = B（e z ）,B（ζ） = g（ζ） ＋ p j=1 b ？j ζ ？j ,g（ζ） is a transcendental entire function of lower order no more than 1/2,and p is an odd positive integer,are studied.It is shown that every non-trivial solution of above equation satisfies the exponent of convergence of zeros equals to infinity.     

复振荡理论中关于超级的角域分布

设f_1和f_2是微分方程f″+A(z)f=0的两个线性无关的解,其中A(z)是无穷级整函数且超级σ_2(A)=0．令E=f_1f_2．本文研究了微分方程f″+A(z)f=0的解在角域中的零点分布,得出E的超级为+∞的Borel方向与零点聚值线的关系．     

On an iterative method for simultaneous inclusion of polynomial complex zeros

Starting from disjoint discs which contain polynomial complex zeros, the iterative interval method of the third order for the simultaneous finding inclusive discs for complex zeros is formulated. The Lagrangean interpolation formula and complex circular arithmetic are used. The convergence theorem and the conditions for convergence are considered. The proposed method has been applied for solving an algebraic equation.     

The Separation of Zeros of Solutions of Higher Order Linear Differential Equations With Entire Coefficients

Suppose that a homogeneous linear differential equation has entire coefficients of finite order and a fundamental set of solutions each having zeros with finite exponent of convergence. Upper bounds are given for the number of zeros of these solutions in small discs in a neighbourhood of infinity.     

Electrostatic models for zeros of polynomials: Old,new, and some open problems

We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity. The leading role is played by the differential equation satisfied by these polynomials. Some new developments, applications and open problems are presented.     

On the zeros of the Scorer functions

Asymptotic approximations are developed for zeros of the solutions Gi(z) and Hi(z) of the inhomogeneous Airy differential equation . The solutions are also called Scorer functions. Tables are given with numerical values of the zeros.     

The Distribution of Generalized Zeros of Solutions of Delay Differential Equations on Time Scales1

In this paper, we consider the following equation where t is on a time scale T. We obtain the distribution of zeros of oscillatory solutions of the above equation.     

A nonlinear eigenvalue problem

We reduce the problem of the zeros of an entire function related to the Painlevé I equation to the eigenvalue problem for the Dirichlet problem for a fourth-order bilinear equation. In a simplified case, this problem reduces to the problem of the position of the zeros of theta functions. We then apply the developed general method to the inverse scattering problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 175–187, November, 2008.