加权流形上加权p-Laplace特征值问题的第一特征值下界估计

本文研究了加权流形上加权p-Laplacian特征值问题的第一特征值下界估计的问题.利用余面积公式、Cavalieri原理以及Federer-Fleming定理,获得了由Cheeger常数或等周常数确定的第一特征值的下界估计.     

双挠积复Finsler流形的局部对偶平坦性

设(M₁,F₁)和(M₂,F₂)是两个强凸的复Finsler流形,λ₁和λ₂是乘积流形M=M₁×M₂上的光滑实值函数,双挠积复Finsler流形(M₁×(λ₁,λ₂) M2,F)是在乘积流形上赋予了复Finsler度量F²=λ₁²F₁²+λ₂²F₂²的复Finsler流形.本文给出了双挠积复Finsler流形是局部对偶平坦流形的充要条件.     

存在通道幅相与互耦两种误差的均匀圆环阵流形曲线拟合方法

针对实际装配后均匀圆环阵的阵列流形向量偏离理论值的问题,提出一种利用单声源从不同方位入射阵列时的阵列幅度相位响应拟合阵列流形模型的算法。考虑阵列存在通道幅度相位一致性偏差和阵元间互耦作用,导出阵列幅度相位响应与流形误差参数的关系式,利用互耦矩阵在模态域可与阵列流形分离的特性,将关于误差参数的关系式降次为线性方程组,再联合多个方位对误差参数做最小二乘估计。对于只存在其中一种误差的特例情况,给出了对应高精度、低复杂度的估计方法。最后,利用数值仿真对所提方法的拟合精度进行评估,拟合后的阵列流形误差距离缩减至10-2量级,水池实验数据也验证了算法在实际应用中的可行性。      

Energy Decay for the Cauchy Problem of the Linear Wave Equation of Variable Coefficients with Dissipation

Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.     

Mean Curvature Flow via Convex Functions on Grassmannian Manifolds

Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the v-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the v-function. Under such restrictions, curvature estimates in terms of v-function composed with the Gauss map can be carried out.     

RELAXATION SOLVERS FOR IDEAL MHD EQUATIONS-A REVIEW

We have developed approximate Riemann solvers for ideal MHD equations based on a relaxation approach in [4], [5]. These lead to entropy consistent solutions with good properties like guaranteed positive density. We describe the extension to higher order and multiple space dimensions. Finally we show our implementation of all this into the astrophysics code FLASH.     

SLANT SUBMANIFOLDS OF A RIEMANNIAN PRODUCT MANIFOLD

In this article, the geometry of the slant submanifolds of a Riemannian product manifold is studied. Some necessary and sufficient conditions on slant, bi-slant and semi-slant submanifolds are given. We research fundamental properties of the distributions which are involved in definitions of semi- and bi-slant submanifolds in a Riemannian product manifold.     

A NOTE ON THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS

Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestrari in. These apriori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.     

从历史的角度引入柯西-黎曼方程

通过对欧拉和黎曼两人获得柯西-黎曼方程的历史过程进行比较。可以发现，欧拉所采取的途径对复分析的引导更加自然．     

关于调和映照的一个Liouville型定理

令M、N是完备Remann流形，设M上不存在任何非平凡的有界调和函数，N的截面曲率具有上界其为k＞0.设uM→N是一个调和映照且u(M)∈BR(P)，其中R＝·如果BR(P)位于P的割迹之内，并且μ（M）∩эBR(P)最多只有一个点，则u必是一个常值映照.