In this paper a new research direction in the optimal interpolation theory is introduced, where the classes of functions to be considered are defined on the whole real axis, and a set of permissible interpolating knots is a denumerable set satisfying certain regular conditions.Some basic problems are formulated, and the exact results obtained in recent years are surveyed.Moreover, many unsolved problems have been put forward.

Basic techniques in the classical theory of commutative associative rings A with unity are:(i) The homological characterization of A in the category of A-modules. (ii) The rings of quotients of prime or semjprime rings A,(iii) The localization Ap at a prime ideal P and the structure sheaf on the prime spectrum of A. All these parts of structure theory are closely related to each other in this case.While technique (i) was successfully applied to non-commutative rings, the parts (ii) and (iii) do not allow a satisfying extension to this more general situation. Moreover, in the special cases which admit corresponding constructions, the interplay between the different points usually gets lost.Replacing the cateaory of left A-modules by a suitable subcategory σ[A] of bimodules over an arbitary (non-associative) ring A a natural extension of all three techniques under consideration which preserves relationships known from the classical situation is obtainedPart (i) and (ii) of the resulting theory can be fou

In this paper we will study Dickson polynomials of the first and second kinds over finite fields. For these polynomials we will discuss some known properties, point out some similarities and differences between the two kinds, and most importantly, indicate a number of open problems concerning these polynomials.

Amahashi gives a necessary and sufficient condition for a graph to have a {1,3, … , 2n-1} -factor. Here we give some necessary and sufficient conditions for a graph to contain a parity-factor,and also we obtain a necessary and sufficient condition for a graph to have parity-factor covering.

Let J be a Hamiltonian operator and ut = JδH/δu be an infinite-dimensional integrable Hamiltonian equation. It is shown that under certain broad assumptions the corresponding stationary equation δH/δu = 0, viewing H as a Lagrangian, can be transformed to a classical Hamiltonian systems qi' = (?)h/(?)pi,pi' = -(?)h/(?)qi, (i= I, …, n) , which is Liouville integrable in the sense that it possesses n first integrals hi which are in involution in pairs. Moreover, a constructive method for calculating the integrals hi is proposed. This connection between finite and infinite-dimensional integrable systems paves a way for constructing a large number of Liouville integrable Hamiltonian systems of finite dimensions.

Let R be a right (left) perfect ring and XR a linearly compact module. If R is a right duo ring then XR has finite length, while this is not true for weakly right duo rings.

Suppose M is a nonorientable closed hyperbolic 3-manifold or an orientable closed hyperbolic 3-manifold with odd first Betti number. Then any smooth action of finite cyclic group can be conjugated to preserve the hyperbolic structure. This result supports a main conjecture in 3-manifold theory.

In this paper we study the vortex methods for the Euler equation of incompressible flow with initial and boundary conditions. Second order extrapolation scheme is applied to calculate the motion of vortex blobs near boundary. Under some corresponding conventional hypotheses second order convergence result is proved, which achieves the same precision as that of the pure initial value problems and can be generalized to any higher order without difficulty. The convergence of the vortex-in-cell method where the stream functions are approximated by the finite element method is also proved.

It is the purpose of this paper to prove the existence of invariant tori and subharmo-nic solutions of high frequence for a nonlinear forced oscillator.

In this paper, using sieve method and Iwaniec's mean value method, we prove that for sufficiently large odd number N, equationN=p1+p2+p3,N/3-N0.63651Related Articles

In this paper,C~(1-0)(D,R)denotes the set of the locally Lipschitz continuousfunctions from D to R,For f∈C~(1-0)(D,R)and x∈D,f(x)denotes the Clarke’sgeneralized gradient of f at x. Definition 1 Let D be an open subset of R~n and f∈C~(1-0)(D,R).Suppose thatε:D→R is a positive function.If a function g∈C~1(D,R)and satisfies