On properties of difference polynomials

We study the value distribution of difference polynomials of meromorphic functions, and extend classical theorems of Tumura-Clunie type to difference polynomials. We also consider the value distribution of f (z)f (z + c).     

PERIODIC CHARACTER OF A DIFFERENCE EQUATION WITH MAXIMUM

We study the periodicity of the positive solutions of a class of difference equations with maximum. We prove that every positive solutions of these equations are eventually periodic.     

Difference Scheme for One Type of Nonlinear Parabolic Equation

We consider the following initial-boundary-value problem of nonlinear parabolic equations Let h denote step size of space and τ denote step size of time. We use the following notation We give the difference scheme for the problem (1)-(2)     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR A CLASS OF DELAY DIFFERENCE EQUATION

We propose a class of delay difference equation with piecewise constant nonlinearity. Such a delay difference equation can be regarded as the discrete analog of a differential equation. The convergence of solutions and the existence of asymptotically stable periodic solutions are investigated for such a class of difference equation.     

CONVERGENT AND DIVERGENT SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR DIFFERENCE SYSTEM

In this paper, we are concerned with the existence of convergent or divergent solutions of two-dimensional nonlinear difference system of the form (?)xn+1=anxn+bnf(yn), yn=cnyn-1+dng(xn). We classify their solutions according to asymptotic behavior and give some sufficient and necessary conditions for the existence of solutions of such classes by using the method of the fixed point theorem. We also give an example and show how the results can he applied to certain difference systems.     

ESTIMATES OF N-FUNCTION AND m-FUNCTION OF MEROMORPHIC SOLUTIONS OF SYSTEMS OF COMPLEX DIFFERENCE EQUATIONS

We apply Nevanlinna theory of the value distribution of meromorphic functions to study the properties of Nevanlinna counting function and proximity function of meromorphic solutions of a type of systems of complex difference equations. Our results can give estimates on the proximity function and the counting function of solutions of systems of difference equations. This implies that solutions have a relatively large number of poles. It extend some result concerning difference equations to the systems of difference equations.     

LONG TIME ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF EXPLICIT DIFFERENCE SCHEME FOR SEMILINEAR PARABOLIC EQUATIONS

In this paper we prove that the solution of explicit difference scheme for a class of semilinear parabolic equations converges to the solution of difference schemes for the corresponding nonlinear elliptic equations in H1 norm as t →∞. We get the long time asymptotic behavior of the discrete solutions which is interested in comparing to the case of continuous solutions.     

APPROXIMATE SOLUTION TO THE PARABOLIC EQUATIONS AND THEIR PICTURE DISPLAY ON THE COMPUTER

In this paper, we consider the approximate solution of the type Ⅰ , Ⅲ initial boundary valued problems of the second order linear parabolic partial differential equations. We use a new difference scheme by suitably combining the difference and the basic recursion of elements in the bivariate spline space S21(Δmn(2)) to construct the approximate solutions. We have proved their convengence. And we will give a flow diagraph to display curved surface on a computer, and give an example.     

Convergence of the Explicit Difference Scheme and the Binomial Tree Method for American Options

This paper is concerned with numerical methods for American option pricing. We employ numerical analysis and the notion of viscosity solution to show uniform convergence of the explicit difference scheme and the binomial tree method. We also prove the existence and convergence of the optimal exercise boundaries in the above approximn.tions.     

THE PATHWISE SOLUTION FOR A CLASS OF QUASILINEAR STOCHASTIC EQUATIONS OF EVOLUTION IN BANACH SPACE Ⅲ

This is the third part of the papers with the same title. We will discuss the problem of convergence of the semi-implicit difference scheme for a class of quasilinear SEE, which generalize the Crandall's work to the stochastic case.     

Uniqueness of Difference-Differential Polynomials of Meromorphic Functions

Ukrainian Mathematical Journal - We study the problems of uniqueness of difference-differential polynomials of finite-order meromorphic functions sharing a small function (ignoring multiplicities)...     

Gr?bner bases in difference-differential modules and difference-differential dimension polynomials

In this paper we extend the theory of Gr?bner bases to difference-differential modules and present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module equipped with the natural filtration. We present and verify algorithms for constructing these Gr?bner bases counterparts. To this aim we introduce the concept of “generalized term order” on ℕ ^m ×ℤ ⁿ and on difference-differential modules. Using Gr?bner bases on difference-differential modules we present a direct and algorithmic approach to computing the difference-differential dimension polynomials of a difference-differential module and of a system of linear partial difference-differential equations. This work was supported by the National Natural Science Foundation of China (Grant No. 60473019) and the KLMM (Grant No. 0705)     

On linear spectral transformations and the Laguerre–Hahn class

We study the Christoffel and Geronimus transformations for Laguerre–Hahn orthogonal polynomials on the real line. It is analysed the modification on the corresponding difference-differential equations that characterize the systems of orthogonal polynomials and the consequences for the three-term recurrence relation coefficients.     

Interlacing of the zeros of contiguous hypergeometric functions

It is well-known that hypergeometric functions satisfy first order difference-differential equations (DDEs) with rational coefficients, relating the first derivative of hypergeometric functions with functions of contiguous parameters (with parameters differing by integer numbers). However, maybe it is not so well known that the continuity of the coefficients of these DDEs implies that the real zeros of such contiguous functions are interlaced. Using this property, we explore interlacing properties of hypergeometric and confluent hypergeometric functions (Bessel functions and Hermite, Laguerre and Jacobi polynomials as particular cases).     

Some problems of approximation theory in the spacesL p on the line with power weight

In the spaces L _p on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional.     

几类亚纯函数q移动微差分多项式的零点与Nevanlinna亏量

The first purpose of this paper is to study the properties on some q-shift difference differential polynomials of meromorphic functions,some theorems about the zeros of some q-shift difference-differential polynomials with more general forms are obtained.The second purpose of this paper is to investigate the properties on the Nevanlinna deficiencies for q-shift difference differential monomials of meromorphic functions,we obtain some relations amongδ(∞,f),δ(∞,f′),δ(∞,f(z)ⁿf(qz+c)^mf′(z)),δ(∞,f(qz+c);f′(z))andδ(∞,f(z)ⁿf(qz+c)^m).     

Termination of algorithm for computing relative Gr?bner bases and difference differential dimension polynomials

We introduce the concept of difference-differential degree compatibility on generalized term orders. Then we prove that in the process of the algorithm the polynomials with higher and higher degree would not be produced, if the term orders ‘?’ and ‘?′’ are difference-differential degree compatibility. So we present a condition on the generalized orders and prove that under the condition the algorithm for computing relative Gr?bner bases will terminate. Also the relative Gr?bner bases exist under the condition. Finally, we prove the algorithm for computation of the bivariate dimension polynomials in difference-differential modules terminates.     

Differential inequalities for entire functions of finite degree and rational functions with prescribed poles

We establish new differential inequalities for the entire functions of finite degree with a majorant an entire function without zeros in the lower half-plane, for the entire functions with constraints on zeros and, as a consequence, for the rational functions with prescribed poles. All cases of equality in the main results are found. The estimates obtained generalize and strengthen some inequalities by Bernstein, Gardner, and Govil for entire functions of finite degree; by Smirnov, Aziz, and Shah for algebraic polynomials; and by Borwein and Erdelyi, Aziz and Shah, and the others for rational functions.     

Degenerate difference-differential equations: Algebraic theory

In this paper we consider the algebraic aspects of the theory of degenerate difference-differential equations. It will be shown that the fundamental algebraic concepts to be used are module theoretic. We have to consider similarity of polynomial matrices in one or more indeterminates. In the case of systems with commensurable lags the underlying modules have a simple structure, because the corresponding ring of scalars is the principal ideal domain of polynomials in one indeterminate. This fact makes it possible to prove a structure theorem for degenerate difference-differential equations with commensurable lags. This theorem shows that degenerate systems of this type essentially are trivial in the sense of Henry [15], i.e., the characteristic quasipolynomial is a polynomial. Further it is shown that coordinate transforms with “time lag” play an essential role for the construction of degenerate equations. The power of the method is demonstrated by some examples, some of which are equations with incommensurable lags.     

Sums of entire functions having only real zeros

We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem, we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang Circle Theorem from statistical mechanics. Also, we state several open problems.