Shared Values of Certain Nonlinear Differential Polynomials of Meromorphic Functions

In this paper, by using the idea of truncated counting functions of meromorphic functions, we deal with the problem of uniqueness of the meromorphic functions whose certain nonlinear differential polynomials share one finite nonzero value.

Notes on Zeros of Differential Polynomials

The value distribution of differential polynomials is studied. The results in this paper improve and generalize some previous theorems given by Yang Chungchun （On deficiencies of differential polynomials, Math. Z., 116（1970）, 197- 204）, H. S. Gopalakrishna and S. S. Bhoosnurmath （On distribution of values of differential polynomials, Indian 3. Pure Appl. Math., 17（1986）, 367-372）, I. Lahiri （A note on distribution of nonhomogeneous differential polynomials, Hokkaido Math. J., 31（2002）, 453-458） and Yi Hongxun （On zeros of differential polynomials, Adv. in Math., 18（1989）, 335-351） et al. Examples show that the results in this paper are sharu.

DIFFERENTIAL POLYNOMIALS SHARING ONE VALUE

In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k ＋ 12. If f^n＋ af^（k）and g^n＋ ag^（k）share b CM and the b-points of f^n＋ af^（k）are not the zeros of f and g, then f and g are either equal or closely related.

On Homogeneous Differential Polynomials of Meromorphic Functions

In this paper, we study one conjecture proposed by W. Bergweiler and showthat any transcendental meromorphic functions f(z) have the form exp(αz + β) if f(z)f″(z) —a(f′(z))~2 7≠ 0, where a ≠ 1, (n±1)/n, n ∈ N. Moreover, an analogous normality criterion isobtained.

Quasi-normal Family of Meromorphic Functions Whose Certain Type of Differential Polynomials Have No Zeros

Define the differential operators φ_(n) for n∈N inductively by φ_(1)[f](z)=f(z)and φ_(n+1)[f](z)=f(z)φ_(n)[f](z)+d/daφ_(n)[f](z).For a positive integer k≥2 and a positive number δ,let F be the family of functions f meromorphic on domain D■C such that φ_(k)[f](z)≠0 and|Res(f,a)-j|≥δ for all j∈{0,1,...,k-1}and all simple poles α of f in D.Then F is quasi-normal on D of order 1.

Differential Polynomials and Normal Families

This paper continues the researches of Yang Lo and others, and gives a further generalization of Gu’s criterion for the normality of a family of meromorphic functions.

Value-Sharing of Meromorphic Functions and Differential Polynomials

This paper is devoted to studying the relationship between meromorphic functions f（z） and g（z） when their differential polynomials satisfy sharing condition weaker than sharing one value IM.

HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS

In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.

Singular Directions and Differential Polynomials of Meromorphic Functions

We prove the existance of a kind of singular directions concerning the differentialpolynomials f（k）+sum from j=0 to k-1 ajfj-afn

The Existence of Periodic Solutions of a Class of <i>n</i>-Degree Polynomial Differential Equations*

This paper deals with a class of n-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (n is an odd number) or two (n is an even number) periodic solutions of the equation is obtained. These conclusions have certain application value for judging the existence of periodic solutions of polynomial differential equations with only one higher-order term.

Numerical Solution for Fractional Partial Differential Equation with Bernstein Polynomials

A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.

SIMULTANEOUSE APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVES BY LAGRANGE INTERPOLATING POLYNOMIALS

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).

ON SIMULTANEOUS APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVE BY INVERSE PAL-TYPE INTERPOLATION POLYNOMIALS

Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +

NORMAL FAMILY OF MEROMORPHIC FUNCTIONS SHARING HOLOMORPHIC FUNCTIONS AND THE CONVERSE OF THE BLOCH PRINCIPLE

In 1996, C. C. Yang and P. C. Hu [8] showed that： Let f be a transcendental meromorphic function on the complex plane, and a ≠ 0 be a complex number; then assume that n 〉 2, n1,… , nk are nonnegative integers such that n1＋… ＋ nk ≥1; thus fn（f′）n1…（f（k））nk-a has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥2. Namely, we prove that fn（f′）n1…（f（k））nk-a（z） has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k≥ 2, and a（z） 0 is a small function of f and n ≥ 2, n1,… ,nk are nonnegative integers satisfying n1＋ …＋ nk ≥1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by d. Yunbo and G. Zongsheng [6], and extension of some problems studied X. Wu and Y. Xu [10]. The main result of this article also leads to a counterexample to the converse of Bloeh＇s principle.

Generalization of Uniqueness of Meromorphic Functions Sharing Fixed Point

In this paper, we study the uniqueness problems of entire and meromorphic functions concerning differential polynomials sharing fixed point and obtain some results which generalize the results due to Subhas S. Bhoosnurmath and Veena L. Pujari [1].

INTEGRABILITY VIA INVARIANT ALGEBRAIC CURVES FOR PLANAR POLYNOMIALDIFFERENTIAL SYSTEMS

We present an introduction to the Darboux integrability theory of planar complex and real polynomial differential systems containing some improvements to the classical theory.

The Center Problem and Time-Reversibility with Respect to a Quadratic Involution for a Class of Polynomial Differential Systems with Order 2 or 3

Most studies of the time-reversibility are limited to a linear or an affine involution.In this paper,the authors consider the case of a quadratic involution.For a polynomial differential system with a linear part in the standard form(-y,x)in R~2,by using the method of regular chains in a computer algebraic system,the computational procedure for finding the necessary and sufficient conditions of the system to be time-reversible with respect to a quadratic involution is given.When the system is quadratic,the necessary and sufficient conditions can be completely obtained by this procedure.For some cubic systems,the necessary and sufficient conditions for these systems to be time-reversible with respect to a quadratic involution are also obtained.These conditions can guarantee the corresponding systems to have a center.Meanwhile,a property of a center-focus system is discovered that if the system is time-reversible with respect to a quadratic involution,then its phase diagram is symmetric about a parabola.

Wave field in two-phase media by the convolutional differentiator method

This paper applies the convolutional differentiator method, based on generalized Forsyte orthogonal polynomial （CFPD）, to simulate the seismic wave propagation in two-phase media. From the numerical results we can see that three types of waves, fast P-waves, S-waves and slow P-waves, can be observed in the seismic wave field. The experiments on anisotropic models demonstrate that the wavefront is elliptic instead of circular and S-wave splitting occurs in anisotropic two-phase media. The research has confirmed that the rules of elastic wave propagation in fluid-saturated porous media are controlled by Biot＇s theory. Experiment on a layered fault model shows the wavefield generated by the interface and the fault very well, indicating the effectiveness of CFPD method on the wavefield modeling for real layered media in the Earth. This research has potential applications to the investigation of Earth＇s deep structure and oil/gas exploration.

POLYNOMIAL DIFFERENTIAL SYSTEM WITH DEGENERATE INFINITY

In this paper we show the distribution of critical points at infinity of n- dimensional polynomial differential systems, and give the conditions, under which the system is degenerate at infinity. Also, we discuss the quadratic systems with degenerate infinity, and obtain some similar properties to 2-dimensional quadratic systems.

Normalization of Indexed Differentials by Extending Grobner Basis Theory

It is a fundamental problem to determine the equivalence of indexed differential polynomials in both computer algebra and differential geometry.However,in the literature,there are no general computational theories for this problem.The main reasons are that the ideal generated by the basic syzygies cannot be finitely generated,and it involves eliminations of dummy indices and functions.This paper solves the problem by extending Grobner basis theory.The authors first present a division of the set of elementary indexed differential monomials E■ into disjoint subsets,by defining an equivalence relation on E■ based on Leibniz expansions of monomials.The equivalence relation on E■also induces a division of a Grobner basis of basic syzygies into disjoint subsets.Furthermore,the authors prove that the dummy index numbers of the sim-monomials of the elements in each equivalence class of E■ have upper bounds,and use the upper bounds to construct fundamental restricted rings.Finally,the canonical form of an indexed differential polynomial proves to be the normal form with respect to a subset of the Grobner basis in the fundamental restricted ring.