本刊英文版2011年54卷第6期摘要

On computing Grbner bases in rings of differential operators MA Xiao Dong,SUN Yao&WANG Ding Kang Abstract Insa and Pauer presented a basic theory of Gr¨obner basis for     

Gr?bner bases in difference-differential modules with coefficients in a commutative ring

Insa and Pauer presented a basic theory of Grbner bases for differential operators with coefficients in a commutative ring and an improved version of this result was given by Ma et al.In this paper,we present an algorithmic approach for computing Grbner bases in difference-differential modules with coefficients in a commutative ring.We combine the generalized term order method of Zhou and Winkler with SPoly method of Insa and Pauer to deal with the problem.Our result is a generalization of theories of Insa and Pauer,Ma et al.,Zhou and Winkler and includes them as special cases.     

Invariant G2V algorithm for computing SAGBI-Gröbner bases

Faugère and Rahmany have presented the invariant F5 algorithm to compute SAGBI-Grbner bases of ideals of invariant rings. This algorithm has an incremental structure, and it is based on the matrix version of F5 algorithm to use F5 criterion to remove a part of useless reductions. Although this algorithm is more efficient than the Buchberger-like algorithm, however it does not use all the existing criteria (for an incremental structure) to detect superfluous reductions. In this paper, we consider a new algorithm, namely, invariant G2V algorithm, to compute SAGBI-Grbner bases of ideals of invariant rings using more criteria. This algorithm has a new structure and it is based on the G2V algorithm; a variant of the F5 algorithm to compute Grbner bases. We have implemented our new algorithm in Maple , and we give experimental comparison, via some examples, of performance of this algorithm with the invariant F5 algorithm.     

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 construct-ing these Grbner bases counterparts. To this aim we introduce the concept of "generalized term order" on Nm ×Zn and on difference-differential modules. Using Grbner bases on difference-differential mod-ules 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.     

Computing polynomial univariate representations of zero-dimensional ideals by Grbner basis

Rational Univariate Representation(RUR) of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively.In 1999,Roullier proposed an efficient algorithm to compute RUR of zero-dimensional ideals.In this paper,we will present a new algorithm to compute Polynomial Univariate Representation(PUR) of zero-dimensional ideals.The new algorithm is based on some interesting properties of Grbner basis.The new algorithm also provides a method for testing separating elements.     

A Generalized Lyapunov-Sylvester Computational Method for Numerical Solutions of NLS Equation with Singular Potential

In the present paper a numerical method is developed to approximate the solution of two-dimensional Nonlinear Schrdinger equation in the presence of a singular potential. The method leads to generalized Lyapunov-Sylvester algebraic operators that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and stable using the Lyapunov criterion, lax equivalence theorem and the properties of the generalized Lyapunov-Sylvester operators.     

Pseudo-differential operators on Sobolev and Lipschitz spaces

In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.     

Anti-commutative Grbner-Shirshov basis of a free Lie algebra

The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Grbner-Shirshov basis for a free Lie algebra. As an application, by using the ...     

Hermitian generalization of the Rarita-Schwinger operators

We introduce two new linear differential operators which are invariant with respect to the unitary group SU（n）. They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator 6z_ and 6z_ and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.     

Double Traveling Wave Solutions of the Coupled Nonlinear Klein-Gordon Equations and the Coupled Schrdinger-Boussinesq Equation

The new multiple(G′/G)-expansion method is proposed in this paper to seek the exact double traveling wave solutions of nonlinear partial differential equations.With the aid of symbolic computation,this new method is applied to construct double traveling wave solutions of the coupled nonlinear Klein-Gordon equations and the coupled Schrdinger-Boussinesq equation.As a result,abundant double traveling wave solutions including double hyperbolic tangent function solutions,double tangent function solutions,double rational solutions,and a series of complexiton solutions of these two equations are obtained via this new method.The new multiple(G′/G)-expansion method not only gets new exact solutions of equations directly and effectively,but also expands the scope of the solution.This new method has a very wide range of application for the study of nonlinear partial differential equations.     

On spinors

For a 2^n-dimensional complex Hermitian vector space S, we prove that any unitary basis of S can be explained as an augmented spinor structure on S. By using this explanation, a SpinC（2n）- action on S is equivalent to an action on a subset of augmented spinor structures. The latter action is a little easy to be understood, and is shown in the last part of this paper. Such kind of understanding could be of use to the discussions of Hermitian manifolds and spin manifolds, especially could help to find connections and elliptical operators.     

Banach空间中关于(H,η)-增生算子的一类新变分包含组问题(英文)

In this paper,we first introduce a new class of generalized accretive operators named(H,η)-accretive in Banach space.By studying the properties of(H,η)-accretive,we extend the concept of resolvent operators associated with m-accretive operators to the new (H,η)-accretive operators.In terms of the new resolvent operator technique,we prove the existence and uniqueness of solutions for this new system of variational inclusions.We also construct a new algorithm for approximating the solution of this system a...     

PERIODIC SOLUTIONS TO NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

A new and convenient method is used to study the existence of periodic solutions to neutral functional differential equations with infinite delay. A new criterion for the existence of periodic solutions is obtained in this paper.     

Continuity in weak topology: First order linear systems of ODE

In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L^p topologies, but also with respect to the weak topologies of the Lp spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way.     

Banach空间中含$(H,\eta)$单调算子变分包含组解的迭代法

In this paper, we introduce and study a new system of variational inclusions involving （H, η）-monotone operators in Banach space. Using the resolveut operator associated with （H, η）- monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the iterative sequence generated by the algorithm.     

Gröbner-Shirshov Bases of Irreducible Modules of the Quantum Group of Type G2

First, the authors give a Grbner-Shirshov basis of the finite-dimensional irreducible module Vq(λ) of the Drinfeld-Jimbo quantum group U_q(G_2) by using the double free module method and the known Grbner-Shirshov basis of U_q(G_2). Then, by specializing a suitable version of U_q(G_2) at q = 1, they get a Grbner-Shirshov basis of the universal enveloping algebra U(G_2) of the simple Lie algebra of type G_2 and the finite-dimensional irreducible U(G_2)-module V(λ).     

ALMOST AUTOMORPHIC MILD SOLUTIONS TO SOME FRACTIONAL DELAY DIFFERENTIAL EQUATIONS

In this paper,a new and general existence and uniqueness theorem of almost automorphic mild solutions is obtained for some fractional delay differential equations,using sectorial operators and the Banach contraction principle.     

关于单调变分不等式的不精确邻近点算法的收敛性分析

We consider a proximal point algorithm(PPA) for solving monotone variational inequalities. PPA generates a sequence by solving a sequence of strongly monotone subproblems .However,solving the subproblems is either expensive or impossible. Some inexact proximal point algorithms(IPPA) have been developed in many literatures. In this paper, we present a criterion for approximately solving subproblems. It only needs one simple additional work on the basis of original algorithm, and the convergence criterion becomes milder. We show that this method converges globally under new criterion provided that the solution set of the problem is nonempty.     

Gr?bner–Shirshov Basis of Derived Hall Algebra of Type A_n

We know that in Ringel–Hall algebra of Dynkin type, the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Gr?bner–Shirshov basis,and the corresponding irreducible elements forms a PBW type basis of the Ringel–Hall algebra. We aim to generalize this result to the derived Hall algebra DH(A_n) of type A_n. First, we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category D~b(A_n) using the Auslander–Reiten quiver of D~b(A_n), and then we prove that all possible compositions between these skew commutator relations are trivial. As an application, we give a PBW type basis of DH(A_n).     

On the Compactness of Grunsky Differential Operators

Grunsky operators play an important role in classical geometric function theory and in the study of Teichmüller spaces. The Grunsky map is known to be holomorphic on the universal Teichmüller space. In this paper the authors deal with the compactness of a Grunsky differential operator. They will give upper and lower estimates of the essential norm of a Grunsky differential operator and discuss when a Grunsky differential operator is a p-Schatten class operator.