LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

On the Spectrum of a Class of Differential Operators and Embedding Theorems

The author considers the embedding problem of weighted Sobolev spaces Hnp in weighted Ls spaces Ls,r,and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions.The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator,and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete.

On a Class of Second Order Partial Differential Operators of Non-principal Type

§1.IntroductionThis paper deals with linear partial differential operators with real principalsymbol.Let P(x,D)be such an operator of mth order with C~∞ coefficients definedin an open subset Ω of R^n and p_m(x,ξ)be its principal symbol.According to thedefinition given by Duistermaat and Hmander(see[1]),P(x,D)is called ofprincipal type at x^0 ∈Ω if for any ξ∈R^n\0 satisfying p_m(x^0,ξ)=0,x=x^0 is not theprojection in Ω of the bicharacteristic strip of P(x,D)through(x^0,ξ).Under thiscondition,they proved that there exists a neighborhood U of x^0,U,such that forany real number s,

TYPES AND CRITERIA OF NONOSCILLATORY SOLUTIONS FOR A CLASS OF SECOND ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONPOSITIVE COEFFICIENTS

In this paper we discuss the types and criteria of nonoscillatory solutions for the fol-lowing second order neutral functional differential equation with nonpositive coefficients

Love in a Changing Society of the Past Half a Century： Suzhou River as a Historical Allegory

Suzhou River, a 2000 film directed by Lou Ye, explores several tragic love stories set in Shanghai around the transitional period of 1980s and 1990s. Many critics have praised its technical excellence, yet generally they have not paid sufficient attention to its subject matter. This paper departs from previous interpretations of the film, which have tended to be premised on superficial readings of the plotline, and contends that the work constitutes a poignant socio-political critique, which is conveyed through the construction of differing love stories set against a changing socio-cultural landscape. The past and the present incarnations of the cardinal female protagonist--who can be understood as a symbol for the average Chinese （woman）--suggest the fact that the society has transformed dramatically across the three disparate eras of the past half a century; accordingly, the identity of the Chinese also shifts tremendously. In this way, Lou Ye in effect constructs a diachronic re-presentation of the changing social mores and varied cultural ethos in a synchronic structure, which is subject to be read as an ingenious historical allegory.

Weighted Integral of Infinitely Differentiable Multivariate Functions is Exponentially Convergent

We study the problem of a weighted integral of infinitely differentiable mul-tivariate functions defined on the unit cube with the L∞-norm of partial derivative of all orders bounded by 1.We consider the algorithms that use finitely many function values as information(called standard information).On the one hand,we obtained that the interpolatory quadratures based on the extended Chebyshev nodes of the second kind have almost the same quadrature weights.On the other hand,by using the Smolyak al-gorithm with the above interpolatory quadratures,we proved that the weighted integral problem is of exponential convergence in the worst case setting.

linear and nonlinear fractional differential equation modified Riemann–Liouville derivatives exact solutions fractional auxiliary sub-equation expansion method Mittag–Leffler function method

In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Kd V equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the（3＋1）-spacetime fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.