Finite Blaschke product interpolation on the closed unit disc

We show how to construct all finite Blaschke product solutions and the minimal scaled Blaschke product solution to the Nevanlinna-Pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data. Based on a result of Jones and Ruscheweyh we note that there always exists a finite Blaschke product of degree at most n−1 that maps n distinct points in the closed unit disc, of which at least one is on the unit circle, into n arbitrary points in the closed unit disc, provided that the points inside the unit circle form a positive semi-definite Pick matrix of full rank. Finally, we discuss a numerical limiting procedure.     

On the Bernstein Inequality for Rational Functions with a Prescribed Zero

We extend some results of Giroux and Rahman (Trans. Amer. Math. Soc.193(1974), 67–98) for Bernstein-type inequalities on the unit circle for polynomials with a prescribed zero atz=1 to those for rational functions. These results improve the Bernstein-type inequalities for rational functions. The sharpness of these inequalities is also established. Our approach makes use of the Malmquist–Walsh system of orthogonal rational functions on the unit circle associated with the Lebesgue measure.     

Projective Blaschke manifolds

In this paper we define the concept of projective Blaschke manifolds and extend the theory of equiaffine differential geometry to the projective Blaschke manifolds. partially supported by NSFC 10771146 and RFDP(20060610004)     

SUBMANIFOLDS IN S~(n+p) WITH PARALLEL MBIUS FORM

§ 1　IntroductionLet x:M→ Sn+p be an n-dimensional submanifold in the unit sphere without umbilicpoints.Let { ei} be a local orthonormal basis with respectto the firstfundamental form I=dx·dx with dual basis{ θi} .Let II =∑ijαhαijθiθjeαbe the second fundamental form of xand H =∑αHαeαbe the mean curvature vector of x,where we use the range of indices:1≤ i,j,k,l≤ n,　　 n + 1≤α,β,γ≤ n + p,Hα=1n∑ihαij,{ eα} is a local orthonormal basis forthe normal bundle ofx.We defi…     

Fundamental solution and single‐chain properties of polylactides

Polylactides (PLA) have been known for several decades and are recently of considerable commercial significance. However, the literature on basic chain properties and solution characterization is divided and inconsistent. In this study, a comprehensive and well‐controlled set of experiments is combined with rigorous quantitative analysis to resolve existing apparent contradictions. Homopolymers and copolymers spanning wide ranges of molecular weight and stereoisomer proportions were prepared by ring‐opening polymerizations of L ‐ and D ‐lactides using stannous octanoate as the catalyst. Samples were characterized by means of: (1) dilute‐solution viscometry in three different solvents; (2) size exclusion chromatography in tetrahydrofuran (THF) with light scattering detection; (3) static multiangle light scattering in a mixed acetonitrile–dichloromethane solvent; (4) variable‐angle spectroscopic ellipsometry; and (5) melt rheology. The data imply that PLA are typical linear flexible polymers; unperturbed PLA chain dimensions are describable in terms of a characteristic ratio of 6.5 ± 0.9, regardless of stereoisomer content. The Schulz‐Blaschke and Mark‐Houwink constants for dilute PLA solutions in chloroform and in THF are determined. For chloroform at 30°C, the correct values are k_SB = 0.302, K = 0.0131 (mL/g), and a = 0.759, while for THF at 30°C, the correct values are k_SB = 0.289, K = 0.0174 (mL/g), and a = 0.736. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 3100–3111, 2005     

Commuting finite Blaschke products with no fixed points in the unit disk

In this paper we study when two finite Blaschke products commute. We complete previous results by Chalendar and Mortini (when they have a fixed point in the unit disk) and by Arteaga (when they do not have a fixed point in the unit disk).     

S~n中子流形的Moebius特性

研究Sn中不舍脐点且Moebius形式为零的子流形的Moebius特性．首先得到子流形的Moebius标准数量曲率与截面曲率的一个关系定理,然后分别利用迹为零的Blaschke张量、Moebius标准数量曲率、截面曲率所满足的某种内蕴关系刻画了驴中子流形的Moebius特性．     

单位球面S~6中的Blaschke等参超曲面

单位球面中的一个无脐点浸入子流形称为Blaschke等参子流形如果它的Mbius形式恒为零并且所有的Blaschke特征值均为常数.维数m4的Blaschke等参超曲面已经有了完全的分类.截止目前,Mbius等参超曲面的所有已知例子都是Blaschke等参的.另一方面,确实存在许多不是Mbius等参的Blaschke等参超曲面,它们都具有不超过两个的不同Blaschke特征值.在已有分类定理的基础上,本文对于5维Blaschke等参超曲面进行了完全的分类.特别地,我们证明了S6中具有多于两个不同Blaschke特征值的Blaschke等参超曲面一定是Mbius等参的,给出了此前一个问题的部分解答.     

Poncelet's theorem, Sendov's conjecture, and Blaschke products

By connecting Blaschke products, unitary dilations of matrices, numerical range, Poncelet's theorem and interpolation, we extend and simplify Gau and Wu's work (Gau and Wu (2004) [10]). We look at Blaschke products' role in the Sendov conjecture.