关于正交张量的向量和反称张量表示

§1 引言 1943年,A.Signorini在建立有限变形的协调条件时,首先采用了有限转动张量的向量表示(见[1,2]),虽然有限转动张量的向量表示已经是一个古老的结果,但它的推导方式始终吸引着数学家和物理学家们的兴趣[3-13].有限转动张量只是一般正交张量的特殊情形:行列式等于+1。[14]首先给出了正交张量向量表示的统一推导。其后[15]对推导作了改进,并给出进一步的系统结果。文献[14,15]的推导要点在于应用Laplace恒等式及张量第一不变量的一个公式。本工作将从另一角度给出定理的证明以及一些其它的结果。     

n维空间中点到直线的距离公式

给出n维空间中直线的定义,借助线性方程组解的理论给出n维空间直线的三种表示形式;定义点在直线上的投影,在此基础上定义点到直线的距离;运用矩阵代数中解线性方程组的方法,先得到原点到直线的距离公式,然后利用坐标变换得到n维空间中点到直线的距离公式,二维、三维空间中点到直线的距离公式是其特例.     

n维空间的Peano曲线

那汤松在文献[1]里,叙述了 Hilbert 关于平面上的 Peano 曲线的构造,并向读者提出了几个问题,其中第四个问题是:“构造三维空间的 Peano 曲线,即在给定的区间[0,1]上,构造这样的三个连续函数(?)(t)、(?)(t)、(?)(t),使所有的点((?)(t),(?)(t),(?)(t))的集合与立方体[0,1]×[0,1]×[0,1]重合”.本文将更一般地、解析地给出在 Jordan 意义下的 n 维欧氏空间的 Peano 曲线 (n≥2),即在区间 [0,1]上,给出 n 个连续函数x_1(t),x_2(t),…,x_n(t),使所有的点 (x_1(t),x_2(t),…,x_n(t))的集合与 n 维立方体[0,1]×[0,1]×…×[0,1] 重合.首先,在区间[0,2]上定义函数     

QDB-张量和SQDB-张量

在文中,我们提出了四类新的结构张量: QDB(QDB0)-张量和SQDB(SQDB0)-张量,并证明了对称偶数阶的QDB-张量和SQDB-张量的正定性,对称偶数阶的QDB0-张量和SQDB0-张量的半正定性.     

n维空间奇点的拓扑分类

在n维相空间内,奇点邻域内的微分方程积分曲线分布的拓扑分类问题,是B.B。在1952年提出来的。其实,这个问题,在H.Poincare的名著[2]中,就已经看到了。大家知道,H.Poincare把主要的平面奇点,分为结点、鞍点、焦点和中心四类;而在空间奇点中,主要增加了鞍焦点一类。他并且指出,四维空间的奇点,主要有八类;五维空间,则有十类。H.Poincare并且把三维空间的非孤立寄点分为结点弧、鞍点弧     

非对称自变量线性各向同性张量函数的表示

本文给出了非对称自变量的线性各向同性标量值和张量值张量函数表示定理的数学证明.     

新的一类三变量正交多项式及其递推公式

研究一类新的三变量正交多项式, 定义为二阶偏微分算子的本征函数, 且在一曲四面体域上正交. 该曲四面体可由普通的四面体映射而得, 可视为 二维Steiner区域的三维推广. 所讨论的正交多项式可视为该区域上的Jacobi多项式. 推导了正交多项式的显式递推公式, 证明其所含的正交多项式项数不依赖多项式的总次数, 沿两个复变量z和$\bar z$方向及单个实变量r方向, 递推公式所含的正交多项式项数分别只为5项与7项. 作为3个特例, 详细讨论了三变量的第1类与第2类Chebyshev多项式及Lengendre多项式.     

一类非对称各向同性张量函数导数的不变表示

将Dui和Chen于2004年提出的求解对称各向同性张量函数导数的方法推广到一类满足可交换条件的非对称各向同性张量函数情况,此类函数比以往研究的更具一般性．在有3个不同特征根时,由可交换性引进张量函数相对应的标量函数,进而求得此类非对称各向同性张量函数及其导数的不变表示形式．在2或3重特征根时,利用求极限的办法给出此类张量函数及其导数的表示形式．     

欧氏空间中具有常典则支撑函数的子流形

本文证明了下述结论：欧氏空间中完备连通且具有常典则支撑函数的子流形必为球面子流形或通过原点的线性子空间，从而改进了关于欧氏空间子流形的一个经典结果。     

Canonical isotopies in Euclidean space

LetG(n,k) denote the space (with the compact-open topology) of homeomorphisms ofR ⁿ which are fixed onR ^k. Theorem 1:G(n, n−2) deforms inG(n, 0) toO(2), whereO(2) is the orthogonal group. Corollary 2: (forn=2) the Kneser theorems. Corollary 3: A EuclideanR ⁿ bundleξ ⁿ overS ^m(m≦∞) which contains anR ⁿ−2 subbundleξ ^{n − 2} is isomorphic, as aG(n,0) bundle, to a Whitney sumξ ^{n − 2} ⊕ξ ². Corollary 4: (n−2) stable homeomorphisms ofR ⁿ orS ⁿ are (n−1) stable, hence stable if orientation preserving. Part of this work represents a portion of the author’s Ph.D. thesis at the University of California, Los Angeles, written under the direction of D. S. Gillman and R. C. Kirby.     

Canonical Representations and Overgroups for Hyperboloids

For the hyperboloid \(X = G/H\), where G = SO₀(p, q) and H = SO₀(p, q ? 1), we define canonical representations R_λ,ν λ ∈ ?, ν = 0, 1, as the restrictions to G of representations \(\tilde R\lambda ,\nu\), associated with a cone, of the group \(\tilde G = \operatorname{SO} _0 (p + 1,q)\). They act on functions on the direct product Ω of two spheres of dimensions p ? 1 and q ? 1. The manifold Ω contains two copies of \(X\) as open G-orbits. We explicitly describe the interaction of the Lie operators of the group \({\tilde G}\) in \(\tilde R\lambda ,\nu\) with the Poisson and Fourier transforms associated with the canonical representations. These transforms are operators intertwining the representations R_λ,ν with representations of G associated with a cone.     

Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n–2)-mean curvature integral of its boundary. This question was considered by Santaló in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes.In dimensions n=2 and 3, Santaló proved that the measure of horospheres intersecting a convex domain is also proportional to the (n–2)-mean curvature integral of its boundary.In this paper we show that this analogy does not generalize to higher dimensions. We express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.     

欧氏空间上Co-regular集的表示式

For the computability of co-regular subsets in metric spaces, the properties of the co-regular subsets and several reasonable representations on co-regular sets have been suggested in this paper. As last, the 'weaker or stronger' relations of these representations have been revealed.     

Grothendieck Representations of Categories and Canonical Noncommutative Topologies

From a representation of a category in terms of a class of Grothendieck categories, a canonical topologisation of the objects follows. The noncommutative topology recently introduced in noncommutative geometry is an example of this. Basic results of sheaf theory are derived in this setting.