Kubota公式的注记

本文研究了n维欧式空间中En中r-维平面的密度与(n-1)维球的体积元之间的关系.利用了复合叠加等的方法,得到了n维欧式空间En的凸体和任意n-r维投影空间Ln-r之间均值积分的关系,推广了Kubota公式得到的结果     

Clifford分析中两类函数的Cauchy积分公式及其相关问题

主要研究了两类函数的Cauchy积分公式及其相关问题.首先给出了Clifford分析中右hypergenic函数的Cauchy积分公式,其次研究了右hypergenic函数拟Cauchy型积分的性质,最后给出了Clifford分析中双hypergenic函数的Cauchy积分公式.     

Clifford分析中对偶的k-Hypergenic函数

研究了取值于实Clifford代数空间Cl_(n+1,0)(R)中对偶的k-hypergenic函数.首先,给出了对偶的k-hypergenic函数的一些等价条件,其中包括广义的Cauchy-Riemann方程.其次,给出了对偶的hypergenic函数的Cauchy积分公式,并且应用其证明了(1-n)-hypergenic函数的Cauchy积分公式.最后,证明了对偶的hypergenic函数的Cauchy积分公式右端的积分是U\Ω_2中对偶的hypergenic函数.     

旋量值函数的Plemelj公式

对比于多复变中的Bochner-Martinelli型积分的Plernelj公式,定义了艾米尔特Clifford分析中旋量值函数的Cauchy型积分及Cauchy主值积分,得到了旋量值函数的Plemelj公式,最后给出一些特殊情形的Bochner-Martinelli型积分的Plemelj公式.     

Clifford分析中k-正则函数的一些性质和高阶Cauchy型积分的边值特性

本文首先给出了定义于R~n取值于Clifford代数C(V_(n,0))中k-正则函数的若干性质,如唯一性定理,Cauchy-Pompeiu公式,高阶Cauchy积分公式,平均值定理等,然后在k-正则函数的高阶Cauchy积分公式的基础上,相应的定义了r次连续可微函数的高阶Cauchy型积分,并给出了它的Cauchy主值,Plemelj公式,边值的Ho|¨lder连续性及其Privalov定理.     

泛Clifford分析中无界域上的Cauchy积分公式和Cauchy-Pompeiu公式

本文研究了泛Clifford分析中的Cauchy积分公式和Cauchy-Pompeiu公式.通过引入修正的Cauchy核,得出了取值在泛Clifford代数上的两公式在无界域上的表达式.此两公式是有界域上的相应结果的推广,并为研究无界域上的边值问题打下了基础.     

超空间中次正则函数的Cauchy积分公式*

在本文中, 首先给出了超空间中次正则函数(sandwich方程 D_xfD_x=0的解)的一些性质, 然后证明了超空间中的Cauchy-Pompeiu公式, 最后得到了超空间中的Cauchy积分公式和Cauchy积分定理.     

Singular Integrals in Several Complex Variables (I)—Henkin Integrals of Strictly Pseudoconvex Domain

对于多复变数强拟凸域的Henkin-Ramirez核或Stein-Kerzman核所定义的Cauchy型积分，本文指出：可以有多种形式的Plemelj公式，甚至Cauchy型积分的极限值可以等于某种Cauchy主值，这些都显示了多复变数函数与单复变数函数本质上的不同。     

四元数分析中k-左正则函数的性质及其Riemann边值问题

讨论了四元数分析中k-左正则函数的若干函数论性质,如Cauchy-Pompeiu公式,Cauchy公式,k-左正则函数的表示,Plemelj公式等.同时考虑了k-左正则函数的Riemann边值问题,通过k-左正则函数的Plemelj公式,将问题转化为奇异积分方程组,再利用积分方程理论和压缩映像原理证明了该问题解的存在唯一性.     

Clifford分析中无界域上Isotonic函数的Plemelj公式

定义了无界域上Isotonic函数的Cauchy型积分和Cauchy主值积分,考虑了其边界性质,得到了无界域上Isotonic函数的Plemelj公式.     

A Generalization of Intersection Formulae of Integral Geometry

We establish extensions of the Crofton formula and, under some restrictions, of the principal kinematic formula of integral geometry from curvature measures to generalized curvature measures of convex bodies. We also treat versions for finite unions of convex bodies. As a consequence, we get a new intuitive interpretation of the area measures of Aleksandrov and Fenchel–Jessen.     

Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities

We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by α-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa–Leindler and Brascamp–Lieb inequalities. Further issues that we transpose to this functional setting are integral-geometric formulae of Cauchy–Kubota type, valuation property and isoperimetric/Urysohn-like inequalities.     

利用插值法证明推广的柯西中值定理

借助插值的思想 ,首先给出函数 f( x)的泰勒公式的行列式表达式 ,推广了柯西中值定理 .据此拉格朗日中值定理、泰勒公式、罗必塔法则均是该结论的推论 ,从而对经典的中值定理、泰勒公式、罗必塔法则给出了统一证明     

Cauchy-type integral formula and Plemelj formula of hypermonogenic functions on unbounded domains

In this paper, we discuss the Cauchy-type integral formula of hypermonogenic functions on unbounded domains in real Clifford analysis, then we extend the Plemelj formula and Cauchy–Pompeiu formula of hypermonogenic functions on bounded domains to unbounded domains. We also deal with the Green-type formula on unbounded domains and get several important corollaries.     

经典Hadamard不等式的高维推广

在n维Euclid空间利用多重积分的一般Stokes公式,将一元凸函数的经典H adam ard不等式在高维空间一般凸区域上进行了推广,得到了相应的高维Hadamard型不等式.这个结果蕴涵了经典的H adam ard不等式以及几个特殊凸体上的H adam ard型不等式.     

Local Tensor Valuations

The local Minkowski tensors are valuations on the space of convex bodies in Euclidean space with values in a space of tensor measures. They generalize at the same time the intrinsic volumes, the curvature measures and the isometry covariant Minkowski tensors that were introduced by McMullen and characterized by Alesker. In analogy to the characterization theorems of Hadwiger and Alesker, we give here a complete classification of all locally defined tensor measures on convex bodies that share with the local Minkowski tensors the basic geometric properties of isometry covariance and weak continuity.     

Local tensor valuations on convex polytopes

Local versions of the Minkowski tensors of convex bodies in $n$ -dimensional Euclidean space are introduced. An extension of Hadwiger’s characterization theorem for the intrinsic volumes, due to Alesker, states that the continuous, isometry covariant valuations on the space of convex bodies with values in the vector space of symmetric $p$ -tensors are linear combinations of modified Minkowski tensors. We ask for a local analogue of this characterization, and we prove a classification result for local tensor valuations on polytopes, without a continuity assumption.     

Application of the fundamental principle to complex Cauchy problem

In this paper we give an explicit formula for the solution of the non-homogeneous complex Cauchy problem with Cauchy data given on a bounded smooth strictly convex domain in a non-characteristic hyperplane. These formulas are obtained using the explicit version of the fundamental principle given in terms of residue currents; moreover, we characterize the domain of definition of the solution and we generalize these techniques to the non-homogeneous Goursat problem.