Analysis of subdivision schemes for nets of functions by proximity and controllability

In this paper we develop tools for the analysis of net subdivision schemes, schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. Sufficient conditions for the convergence of such a sequence of refined nets, and for the smoothness of the limit function, are derived in terms of proximity to a bivariate linear subdivision scheme refining points, under conditions controlling some aspects of the univariate functions of the generated nets. Approximation orders of net subdivision schemes, which are in proximity with positive schemes refining points are also derived. The paper concludes with the construction of a family of blending spline-type net subdivision schemes, and with their analysis by the tools presented in the paper. This family is a new example of net subdivision schemes generating C¹ limits with approximation order 2.     

Subdivision rules and virtual endomorphisms

Suppose f : S ² → S ² is a postcritically finite branched covering without periodic branch points. If f is the subdivision map of a finite subdivision rule with mesh going to zero combinatorially, then the virtual endomorphism on the orbifold fundamental group associated to f is contracting. This is a first step in a program to clarify the relationships among various notions of expansion for noninvertible dynamical systems with branching behavior.      

A novel optical subdivision method for dual-frequency interferometer

The theory of optical subdivision techniques of dual-frequency laser interferometers is stated. And a novel optical subdivision technique is proposed originally to enhance resolution of a commercial interferometer by adding some corner-cubes. Then the performance of the interferometer is tested. The interferometer resolution of 1.24 nm and the average error of below 2 nm are achieved by using the technique. The most novel of the optical subdivision technique is without λ/4 plates. It is less sensitive to environmental changes, it has prodigious potential to improve resolution farther and it can reduce polarization mixing error.     

Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces

Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory.

     

几何造型的有理矩阵细分方法

Ｍｉｃｃｈｅｌｌｉ,Ｐｒａｕｔｚｓｃｈ给出了一类生成曲线的细分法－矩阵细分方程,但该方法仅能生成多项式类型的曲线,为了弥补其不足,本文提出了有理矩阵细分方法,并证明了其生成曲线的优良性质,例如凸包性,几何不变性,变差缩减性等。     

An accessibility theorem for infinite graph minors

We prove the following recent conjecture of Halin. Let Γ₀ be the class of all graphs, and for every ordinal μ > 0 let Γ_μ be the class of all graphs containing infinitely many disjoint connected graphs from Γ_λ, for every λ < μ. Then a graph lies in all these classes Γ_μ if and only if it contains a subdivision of the infinite binary tree. Published by John Wiley & Sons, Inc., 2000 J Graph Theory 35: 273–277, 2000     

二次C-曲线研究

给出具有相同控制顶点的二次C-曲线与二次有理Bézier曲线表示同一参数曲线段的充要条件,由此得到了二次C-曲线不能精确表示双曲线段的结论;另外,还给出了二次C-曲线在任意一点的细分公式.     

三角域上Said-Bézier型广义Ball基的对偶泛函及其应用

A family of Said-Bézier type generalized Ball(SBGB) bases and surfaces with a parameter H over triangular domain is introducermediate surfaces.To convert different bases and surfaces,the dual functionals of bases are presented.As an application of dual functionals,the subdivision formulas for surfaces are established.     

CONVERGENCE RATE OF VECTOR SUBDIVISION SCHEME

In this paper,some characterizations on the convergence rate of both the homoge- neous and nonhomogeneous subdivision schemes in Sobolev space are studied and given.