张量代数

In this paper,we propose a concept called interval implicitization of rational bézier surfaces,that is ,finding an interval algebraic surface with lower degree which bounds a given parametric rational bézier surface such that some objective function involving the width and the tention term of the interval algebraic surface is minimized.

本文提出有理Bézier曲面的区间隐式化的概念，即找到一条较低次的区间代数曲面使得给出的有理Bézier曲面落在该区间代数曲面内，并使得一包含区间代数曲面的宽度和张量项的目标函数达到最小。

In chapter 2, a new concept- property T〓 is introduced, it is proved that nest subalgebras of von Neumann algebras, finite width CSL and atomic CSL subalgebras of von Neumann algebras have property T〓. Some tensor product formulae of nest subalgebras and CSL subalgebras in von Neumann algebras are obtained. Tensor products of W〓-dynamical systems are studied. Finally, the essential commutant of tensor product of operator algebras is discussed.

第二章我们引入了性质T〓这一新概念，并证明了任一vonNeumann中的任何套子代数，有限宽度的可交换格子代数和原子可交换格子代数都具有性质T〓；得到了von Neumann代数中套子代数和可交换格子代数的张量积公式；研究了W*-动力系统的张量积问题；最后讨论了张量积代数的本性换位。

By Llibre's algebraic invariant theory, at first on the basis of classification for quadric and quadric homogeneous binary polynomial, the plane linear and cubic homogeneous polynomial differential systems are classified.

利用Llibre的代数不变式理论，首先由二元二次和二元四次多项式的分类结果，对一次和三次齐次多项式微分系统进行代数分类，同时补充了已有结果中出现的漏洞；其次，由称共变张量空间的性质，对缺二次项的三次微分系统在保证轨线走向不变的前提下进行代数分类，使分类后同类系统的示性多项式有相同零点；最后通过讨论一类简单系统的有界性说明了分类的方便方处。

Finally, after defining an evaluation module of the corresponding Loop superalgebra (Section 4), two major results of the paper -Theorem 4.land Theorem 4.2 are proved: Theorem 4.1 reduces the irreducibility of the tensor product of finitely many evaluation modules to the irreducibility of the tensor product of finitely many irreducible modules of a nilpotent Lie superalgebra; Theorem 4.2 gives a criterion for the tensor product of such modules to be irreducible.

第4节在定义了相应的Loop超代数的赋值模之后，证明了本文的两个主要结论：定理4.1和定理4.2。定理4.1将有限多个赋值模其张量积的不可约性归结为一幂零李超代数的限多个不可约模其张量积的不可约性；定理4.2利用不可约指标给出了一幂零李超代数的限多个不可约模其张量积仍不可约的判别准则。

It is not easy to solve the problem of tensor product multilinear mapping if only matrix algebra knowledge is used. The introduction of structure matrix will make this problem much easier to be solved and more acceptable among people.

在解决张量积多重线性映射问题时，只用矩阵代数知识解决问题有些繁琐，而引入结构矩阵来解决张量积多重线性映射问题，不仅能使问题变得更加简单而且更加容易理解接受。

In tensor algebra the vector appears as a speical case of a more general concept, which includes stress and inertia tensors

在张量代数中，矢量是做为更普遍的概念的一种特殊情况出现的，这种更普遍的概念包括应力张量和应变张量。

In this part, a new method, using tensor algebra, is developed to yield any order approximate analytical solution of a universal nonlinear dynamic system equation, which is abstracted from a large class of nonlinear dynamic systems in many different disciplines.

该篇采用张量代数，首次用严格的分析方法获得了适合不同学科的、普遍的非线性动力系统方程任意阶近似解析解。

Briefly covered vector and tensor notation, vector and tensor algebra, wave kinematics, eigenvectors/values, dimensional analysis, and scale analysis.

简要的介绍向量和张量符号，向量和张量代数，波的运动，特徵向量／值，向量分析，以及尺度分析。

This method is based on tensor and tensor algebra. To utilize the multidimensional data of the remote sensing image, the vector-based learning machine is generalized to the tensor-based learning machine which accepts tensors as input, then the convex optimization theory and the alternating projection procedure are used to get the solution of the TLM.

该方法基于张量数据模型和张量代数运算，针对遥感影像数据多维或高维的特点，将基于向量的监督法学习机扩展为基于张量的监督法学习机，然后利用凸函数最优化理论和交互投影迭代法求得张量学习机的最优解。

Tensor algebra is tied to coordinates.

张量代数则离不开坐标系。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。