SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES

In this paper,we mainly investigate the value distribution of meromorphic functions in Cmwith its partial differential and uniqueness problem on meromorphic functions in Cmand with its k-th total derivative sharing small functions.As an application of the value distribution result,we study the defect relation of a nonconstant solution to the partial differential equation.In particular,we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation.

Extension of covariant derivative(Ⅱ): From flat space to curved space

This paper extends the classical covariant deriva tive to the generalized covariant derivative on curved sur faces. The basement for the extension is similar to the pre vious paper, i.e., the axiom of the covariant form invariabil ity. Based on the generalized covariant derivative, a covari ant differential transformation group with orthogonal duality is set up. Through such orthogonal duality, tensor analy sis on curved surfaces is simplified intensively. Under the covariant differential transformation group, the differential invariabilities and integral invariabilities are constructed on curved surfaces.

General Integral Formulasof （0, q）（q 〉 0）Differential Forms on the Analytic Varieties

In this paper, firstly using different method and technique we derive the cor-responding integral representation formulas of （0, q）（q 〉 0） differential forms for the twotypes of the bounded domains in complex submanifolds with codimension-m. Secondly weobtain the unified integral representation formulas of （0, q）（q 〉 0） differential forms for thegeneral bounded domain in complex submanifold with codimension-m, which include Hatzi-afratis formula, i.e. Koppelman type integral formula for the bounded domain with smoothboundary in analytic varieties. In particular, when m -- 0, we obtain the unified integralrepresentation formulas of （0, q）（q 〉 0） differential forms for general bounded domain in Cn,which are the generalization and the embodiment of Koppelman-Leray formula.

Isotope effect on the stereodynamics for the collision reaction H+LiF(v=0, j=0)→ HF+Li

Stereodynamics for the reaction H＋LiF（v = 0, j = 0） → HF＋Li and its isotopic variants on the ground-state （12A＇） potential energy surface （PES） are studied by employing the quasi-classical trajectory （QCT） method. At a collision energy of 1.0 eV, product rotational angular momentum distributions P（0r）, P（~r）, and P（Or, Cr）, are calculated in the center-of-mass （CM） frame. The results demonstrate that the product rotational angular momentum j＇ is not only aligned along the direction perpendicular to the reagent relative velocity vector k, but also oriented along the negative y axis. The four generalized polarization-dependent differential cross sections （PDDCSs） are also computed. The PDDCS00 distribution shows a preferential forward scattering for the product angular distribution in each of the three isotopic reactions, which indicates that the title collision reaction is a direct reaction mechanism. The isotope effect on the stereodynamics is revealed and discussed in detail.

Implicit Function Theorem and Its Application to a Ulam's Problem for Exact Differential Equations

We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g（x, y） ＋ h（x, y）y＇ =0.

Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras （In Memory of Professor Guenter Krause）

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method： Grobner-Shirshov basis method. We develop the GrSbner-Shirshov basis theory of differential difference al- gebras, and of finitely generated modules over differential difference algebras, respectively. Then, via GrSbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

Closed-form solutions to fractional-order linear differential equations

The definitions and properties of widely used fractional-order derivatives are summarized in this paper.The characteristic polynomials of the fractional-order systems are pseudo-polynomials whose powers of the complex variable are non-integers.This kind of systems can be approximated by high-order integer-order systems,and can be analyzed and designed by the sophisticated integer-order systems methodology.A new closed-form algorithm for fractional-order linear differential equations is proposed based on the definitions of fractional-order derivatives,and the effectiveness of the algorithm is illustrated through examples.

Meshfree Approximation for Stochastic Optimal Control Problems

In this work,we study the gradient projection method for solving a class of stochastic control problems by using a mesh free approximation ap-proach to implement spatial dimension approximation.Our main contribu-tion is to extend the existing gradient projection method to moderate high-dimensional space.The moving least square method and the general radial basis function interpolation method are introduced as showcase methods to demonstrate our computational framework,and rigorous numerical analysis is provided to prove the convergence of our meshfree approximation approach.We also present several numerical experiments to validate the theoretical re-sults of our approach and demonstrate the performance meshfree approxima-tion in solving stochastic optimal control problems.