Pick矩阵负特征值个数有界的函数

The Takagi problem on the disk and bidisk

We give a new proof on the disk that a Pick problem can be solved by a rational function that is unimodular on the unit circle and for which the number of poles inside the disk is no more than the number of non-positive eigenvalues of the Pick matrix. We use this method to find rational solutions to Pick problems on the bidisk.

关于广义Schur函数的一个多点插值问题

On a multi-point interpolation problem for generalized Schur functions

具有多个插值条件的非退化Nevanlinna-Pick-Carathéodory-Fejer插值问题对于每个$κ\ge κ_{\rm min}$在广义Schur类$\cS_κ$中总是有无穷多个解，其中整数$κ_{\rm min}$等于与该问题相关联的Pick矩阵的负特征值的个数，并且完全由插值数据确定.一个线性分式描述的所有$\cS_{κ_{\rm min}}$解决方案的（非退化）问题是众所周知的。在本文中，我们提出了一个类似的结果为任意$κ\ge κ_{\rm min}$。

Laplace本征函数节点几何中的符号和面积

Sign and area in nodal geometry of Laplace eigenfunctions

本文讨论了Laplace-Beltrami算子在闭曲面上的渐近节几何。给定一个对应于一个大的本征值的本征函数f，我们研究了符号（f）的分布相对于表面积的局部不对称性。它的测量方法如下：取一个以节线{f=0}为中心的圆盘，在这个圆盘上随机选取一个点。函数在所选点取正值的概率是多少？我们表明，这个数量可能会衰减的特征值无穷大，但从来没有比这更快。换句话说，可能只出现轻微的局部不对称。证明方法结合了Donnelly-Fergusman和Nadirashvili关于单位圆上调和函数的一个新结果。

On the multipliers at fixed points of self-maps of the projective plane

This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely deter…