非奇异Jordan矩阵最小奇异值的下界定理的两种证明方法

用两种方法证明出非奇异Jordan块的最小奇异值下界,进而得出一般非奇异Jordan阵的类似结论。

k次根式最大下界的两个定理及其应用

1引言关于k次根式(?)的上界早已有解决的方法,文[1]探讨了其特殊情形,二次根式(?)的下界,即以下定理:

对称多胞形的极值性质及其应用

凸多胞形现代理论的主要成就是被称之为Dehn-Sommerville关系的上界定理和下界定理,它们属于凸多胞形的经典组合理论．本文建立了关于对称凸多胞形的两个极值定理,它们可视为凸多胞形度量理论中的上界定理和下界定理,另外给出了两个极值定理的一个应用．

The Estimation for Lower Bounds of the Solutions of Fermat's Equation

Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.

A Simple Proof of the Lower Bounds of Distortion Theorems on the Roper-Suffridge Operator

Liczberski-Starkov firstfound a lower bound for ｜｜D（f）｜｜ near the origin, where f（z）=（F（z1）,√F1（z1）z2,…,√F＇（z1）zn）＇is the Roper-Suffridge operator on the unit ball Bn in Cn and F is a normalized convex function on the unit disk. Later, Liczberski-Starkov and Hamada-Kohr proved the lower bound holds on the whole unit ball using a complex computation. Here we provide a rather short and easy proof for the lower bound. Similarly, when F is a normalized starlike function on the unit disk, a lower bound of ｜｜D（f）｜｜ is obtained again.

Eigenvalue Comparison Theorems on Finsler Manifolds

Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau's estimation of the first eigenvalue is also given.