有理曲线和曲面的分片多项式逼近

本文利用摄动的思想,以摄动有理曲线（曲面）的系数的无穷模作为优化目标,给出了用多项式曲线（曲面）逼近有理曲线（曲面）的一种新方法．同以前的各种方法相比,该方法不仅收敛而且具有更快的收敛速度,并且可以与细分技术相结合,得到有理曲线与曲面的整体光滑、分片多项式的逼近．     

Forelli-Rudin type theorem in pluriharmonic Bergman spaces with small index

It is proved that the Bergman type operatorT, is a bounded projection from the pluriharmonic Bergman spaceL ^p (B)∩h(B) onto Bergman spaceL ^p (B) ∩ H(B) for 0p 1 ands (p¹-1)(n+1). As an application it is shown that the Gleason’s problem can be solved in Bergman space L^P(B)∩H(B) for 0p 1. Project supported by the National Natural Science Foundation of China (Grant No. 19871081) and the Doctoral Program Foundation of the State Education Commission of China.     

Maximally Nonlinear Functions and Bent Functions

We give a construction of bent functions in 2n variables, here n is odd, by using maximally nonlinear functions on GF(2 ⁿ ).     

Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions

A new class of generalized convex set-valued functions, termed nearly-subconvexlike functions, is introduced. This class is a generalization of cone-subconvexlike maps, nearly-convexlike set-valued functions, and preinvex set-valued functions. Properties for the nearly-subconvexlike functions are derived and a theorem of the alternative is proved. A Lagrangian multiplier theorem is established and two scalarization theorems are obtained for vector optimization.     

Asymptotic Uniqueness of Best Rational Approximants of Given Degree to Markov Functions in L 2 of the Circle

We improve over a sufficient condition given in [8] for uniqueness of a nondegenerate critical point in best rational approximation of prescribed degree over the conjugate-symmetric Hardy space of the complement of the disk. The improved condition connects to error estimates in AAK approximation, and is necessary and sufficient when the function to be approximated is of Markov type. For Markov functions whose defining measure satisfies the Szego condition, we combine what precedes with sharp asymptotics in multipoint Padé approximation from [43], [40] in order to prove uniqueness of a critical point when the degree of the approximant goes large. This lends perspective to the uniqueness issue for more general classes of functions defined through Cauchy integrals.     

Monodromy of Rational KZ Equations and Some Related Topics

A special class of integrable Fuchsian systems on C ⁿ related to KZ equations is considered. We survey the construction of such systems and the list of the structural properties their monodromy representations. The relation of the Fuchsian systems obtained by the Veselov construction assosiated with a deformation of the A _n–1-type root system and the Gauss–Manin connection of the natural projection C ⁿ C ^n–1 is described. In this case, we prove that the monodromy representation is equivalent to the Burau representation of the Artin braid group. For a deformations of the other root system, we introduce generalized Burau representations. We conjecture that the integrable Fuchsian systems related to essential new finite sets of the vectors described by Veselov and Chalykh are the result of the Klares–Schlesinger isomonodromic deformations (or transformation) of the integrable Fuchsian system related to the Coxeter root systems.     

Certain Convolution Operators for Meromorphic Functions

Let (p N) be the class of functions analytic in 0 < |z| < 1. A convolution operator L_p(a, c) on p is introduced. This paper gives some sharp inequalities for f(z) satisfying Re{(1 – )z^pL_p(a, c) f(z) + z^pL_p(a + 1, c) f(z)} > , where 0, < 1, a > 0 and c 0, –1, –2,....AMS Subject Classification (1991) 30C45 30A10     

Evaluating Lipschitz Constants for Functions Given by Algorithms

This paper describes an efficient method (O(n)) to evaluate the Lipschitz constant for functions described in some algorithmic language. Considering arithmetical operations as the basis of the algorithmic language and supported by control structures, the rules to evaluate such Lipschitz constants are presented and their correctness is proved. An extension of the method to evaluate Lipschitz constants over interval domains is also presented. Examples are presented, but the effectiveness of the method is doubtful when compared to other approaches, and effective enhancements based on slope evaluations are also explored.