深分层流体中内孤立波的相互作用——Ⅰ.弱相互作用 *

采用Lagrange观点研究分层流体中内孤立波的弱相互作用,它包括不同模式孤立波间的追撞和迎撞,以及相同模式孤立波间的迎撞.分析表明在有限深度情形每个波遵循ILW方程,而在无限深度情形每个波满足Benjamin-Ono方程,相互作用的主要效应体现在相移上.     

例析等积法的解题技巧

初中数学中的等积法是一种十分重要的解题思路与方法,其实质是一种转换思想,例如运用“两个三角形等底等高则面积相等”的性质,把一些较复杂的难以直接解决的问题,转化为较简单的能够间接解决的问题,从而使问题得到简捷的解答.本文中结合四类典型实例,探讨和总结了运用等积法解题的方法与技巧.     

关于局部诺伊曼等周常数（英文）

本文引入流形中一个相对区域上的相对等周不等式,并说明它等价于相对Sobolev不等式.在倍体积假设条件下,推出了度量球上的经典等周不等式.最后,讨论了该不等式在一些曲率条件下的应用和凯勒—里奇流中的应用.     

基于图论的快速图像等周分割算法

等周算法应用于图像分割,因多次求解线性方程组,造成时间复杂度过大.针对于此,通过以区域代替像素、设置新的权值函数、降低迭代次数提出了一种新的快速等周算法,并将其应用于图像分割,取得了很好效果.     

Versal unfolding of equivariant bifurcation problems in more general case under two equivalent groups

For the unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry,the versal unfolding theorem with respect to left-right equivalence is obtained by using the related methods and techniques in the singularity theory of smooth map-germs.The corresponding results in[4,9]can be considered as its special cases.A relationship between the versal unfolding w.r.t.left-right equivalence and the versal deformation w.r.t.contact equivalence is established.     

从积分几何的观点看几何不等式——纪念李国平院士吴新谋教授诞辰100周年

该文先介绍一些中国数学家在几何不等式方面的工作.作者用积分几何中著名的Poincarè公式及Blaschke公式估计一随机凸域包含另一域的包含测度, 得到了经典的等周不等式和Bonnesen -型不等式.还得到了一些诸如对称混合等周不等式、Minkowski -型和Bonnesen -型对称混合等似不等式在内的一些新的几何不等式.最后还研究了Gage -型等周不等式以及Ros -型等周不等式.     

二阶椭圆问题的四边形等参元数值积分的影响

对于曲边区域上二阶椭圆型问题,本文研究四边形等参有限元逼近格式的收敛性.为了便于单元刚度矩阵和荷载向量的计算,构造几种简单的数值积分格式,并提出一个仅有两个积分点的最优的数值积分公式,这是目前为止积分点最少的最优的数值积分公式.     

幂等算子线性组合的幂等性

设P与Q是Hilbert空间中的两个不同的幂等算子.本文主要刻画了幂等算子P与Q的线性组合仍是幂等算子的充要条件,从而推广了Baksalary与Baksalary (2000)的结论.值得指出的是,我们通过严密的推理发现,其定理的条件P_1P_2≠P_2P_1是非必要的.     

幂等矩阵线性组合的可逆性

设T1,T2,T3是三个不同的两两相互可交换的n×n非零的三次幂等矩阵,并且c1,c2,c3是非零数.本文主要给出了线性组合c1T1 c2T2 c3T3可逆性的刻画.     

THE UNFOLDING OF EQUIVARIANT BIFURCATION PROBLEMS WITH PARAMETERS SYMMETRY

In this paper versal unfolding theorem of multiparameter equivaxiant bifurcation problem with parameter symmetry is given. The necessary and sufficient condition that unfolding of multiparameter equivariant bifurcation problem with parameter symmetry factors through another is given. The corresponding results in [1]-[6] are generalized.     

Depth notions for orthogonal regression

Global depth, tangent depth and simplicial depths for classical and orthogonal regression are compared in examples, and properties that are useful for calculations are derived. The robustness of the maximum simplicial depth estimates is shown in examples. Algorithms for the calculation of depths for orthogonal regression are proposed, and tests for multiple regression are transferred to orthogonal regression. These tests are distribution free in the case of bivariate observations. For a particular test problem, the powers of tests that are based on simplicial depth and tangent depth are compared by simulations.     

Depth estimators and tests based on the likelihood principle with application to regression

We investigate depth notions for general models which are derived via the likelihood principle. We show that the so-called likelihood depth for regression in generalized linear models coincides with the regression depth of Rousseeuw and Hubert (J. Amer. Statist. Assoc. 94 (1999) 388) if the dependent observations are appropriately transformed. For deriving tests, the likelihood depth is extended to simplicial likelihood depth. The simplicial likelihood depth is always a U-statistic which is in some cases not degenerated. Since the U-statistic is degenerated in the most cases, we demonstrate that nevertheless the asymptotic distribution of the simplicial likelihood depth and thus asymptotic α-level tests for general types of hypotheses can be derived. The tests are distribution-free. We work out the method for linear and quadratic regression.     

~2样本深度的基本性质

本文讨论由L2深度修正得到的L2深度相应的样本深度的性质,得到了样本深度的相合性和渐近正态性,并证明了它在任意紧集上的一致相合性．最后,基于上述性质简要讨论了样本深度等高的一些性质．     

On data depth in infinite dimensional spaces

The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite dimensional data, there have been some recent attempts to develop depth functions for data in infinite dimensional spaces. In this paper, we consider some notions of depth in infinite dimensional spaces and study their properties under various stochastic models. Our analysis shows that some of the depth functions available in the literature have degenerate behaviour for some commonly used probability distributions in infinite dimensional spaces of sequences and functions. As a consequence, they are not very useful for the analysis of data satisfying such infinite dimensional probability models. However, some modified versions of those depth functions as well as an infinite dimensional extension of the spatial depth do not suffer from such degeneracy and can be conveniently used for analyzing infinite dimensional data.     

一类非线性波动方程位势井深度函数的连续性

本文研究一类非线性波动方程位势井深度函数的连续性.通过引入位势井深度函数并给出其性质,给出了位势井深度函数连续性的证明.而位势井深度函数连续性保证了在其基础上得到的位势井族有意义.     

Second-order accuracy of depth-based bootstrap confidence regions

We consider the problem of setting bootstrap confidence regions for multivariate parameters based on data depth functions. We prove, under mild regularity conditions, that depth-based bootstrap confidence regions are second-order accurate in the sense that their coverage error is of order n⁻¹, given a random sample of size n. The results hold in general for depth functions of types A and D, which cover as special cases the Tukey depth, the majority depth, and the simplicial depth. A simulation study is also provided to investigate empirically the bootstrap confidence regions constructed using these three depth functions.     

统计深度函数及其应用

次序统计量在一维统计数据分析中起着很重要的作用．多年来,人们一直在商维数据处理和分析中寻找“次序统计量”,却没有得到很满意的结果．由于缺少自然而有效的高维数据排序方法,因而象一维“中位数”的概念很难推广到高维．统计深度函数则提供了高维数据排序的一种工具,其主要思想是提供了一种从高维数据中心(最深点)向外的排序方法．不仅如此,统计深度函数已经在探索性高维数据分析,统计判决等方面带给我们一种全新的前景,并在工业、工程、生物医学等诸多领域得到很好的应用．本文介绍了统计深度函数概念及其应用,讨论了位置深度函数的标准,介绍了几种常用的统计深度函数．给出了由深度函数特别是由投影深度函数所诱导的位置和散布阵估计,介绍了它们的诸多优良性质,如极限分布,稳健性和有效性．由于在大多数场合下,高崩溃点的估计不是较有效的估计,而由统计深度函数所诱导的估计具有多元仿射不变性,并能提供理想的稳健性与有效性之间的平衡,本文还讨论了基于深度的统计检验和置信区域,介绍了统计深度函数的其他应用,如多元回归、带有变量误差模型、质量控制等,以及实际计算问题．指出了统计深度函数领域有关进一步的工作和研究方向．     

Depth for complexes, and intersection theorems

This paper introduces a new notion of depth for complexes; it agrees with the classical definition for modules, and coincides with earlier extensions to complexes, whenever those are defined. Techniques are developed leading to a quick proof of an extension of the Improved New Intersection Theorem (this uses Hochster's big Cohen-Macaulay modules), and also a generalization of the “depth formula” for tensor product of modules. Properties of depth for complexes are established, extending the usual properties of depth for modules. Received May 6, 1997; in final form December 3, 1997     

Asymptotics of stationary points of local simplicial depth in the univariate case

Local depth (see Agostinelli and Romanazzi, 2011) extends the usual depth definition so as to account for multimodal distributions and clustered data. Applications include rankings of data according to centrality in suitable neighborhoods and locations of partial centers. Here the basic definitions and properties of local simplicial depth in the univariate case are reviewed. The univariate case is special because the derivative of the depth function has a simplemathematical representation. Using this result and the theory of empirical processes, the asymptotic behavior of the stationary points of local simplicial depth is further investigated with some new findings. Illustrations based on simulated and real data are given.