四元数

In this note, we show that for any two matrices $A$ and $B$ over a generalized quaternion algebra defined on an arbitrary field ${\bf F}$ of characteristic not equal to two, if $A$ and $B$ are similar and the main diagonal elements of $A$ and $B$ are in $\bf F$, then their traces are equal.

本文对于特征不是2的任意域${\bf F}$上定义的广义四元数代数上的两个矩阵$A$和$B$，给出如果$A$和$B$相似并且它们的主对角线上的元素在${\bf F}$中，那么它们的迹相等。

It helps to research the solution of the situation and stability of the real quaternion matrix equation.

由此有助于研究实四元数矩阵方程的解的情况和解的稳定性。

Bernstein type theorem; Quaternion Euclidean space; minimal Lagrangian graphs; Twisted special Lagrangian submanifold; twisted normal bundle; Austere submanifolds; Real space form; Complex space form; Hamiltonian minimal submanifold; A Lagrangian submanifold with conformal Maslov form

基础科学，数学，几何、拓扑伯恩斯坦型定理；四元数欧氏空间；极小拉格朗日图；扭曲特殊拉格朗日子流形；扭曲法丛； Austere子流形；实空间形式；复空间形式；哈密尔顿极小子流形；具有共形Maslov形式的拉格朗日子流形

Computer; mathematical problems; Mizar project; quaternion numbers; algebra structure

计算机；数学问题； Mizar系统；四元数；代数结构

This paper firstly reviews the history of Automated Theorem Proving and Mizar,then gives a description of methods of mechanical theorem proving and verification to mathematical proposition under Mizar system.Based on that basis,the paper has done some Mizar researches on mixed operation of quaternion,differentiati- on of special composite function and the application of Rolle\'s theorem.

本文首先介绍了定理机器证明和Mizar语言系统的发展历史，其次简单描述了Mizar系统下定理机器证明和校验数学命题的方法，在此基础上对四元数的混合运算，特殊复合函数的微分以及洛尔定理的应用等方面进行了Mizar实现。

The existence and forms of the generalized righteigenvalues of real nonsingular quaternion matrix were put forward.

给出了实四元数矩阵正则对的广义右特征值的存在性和表达形式。

In this paper, based on the concept of normal quaternion, a new algorithm for the initial alignment of the integrated system is studied; the alignment algorithm is processed only on the basis of the position and velocity data of GPS.

基于规范化四元数的概念，研究了一种仅利用GPS的位置和速度数据对组合系统进行初始对准的新算法，克服了传统滤波方法进行初始对准存在的诸多局限性。

Vector is a useless survival, or offshoot from quaternions, and has never been of the slightest use to any

向量是无用的幸存物或四元数的无价值的支流，对任何创造，从未有过最微细的应用。

The former is an iterative method, and quaternion was utilized to simplify the matrix orthogonalization procedure.

改进的单点估计算法是一种迭代算法，它计算简单，收敛速度快，同时由于采用四元数表示姿态矩阵，简化了矩阵的正交化过程。

The computational flow of light propagation and the key techniques of visualization are presented. An optical system was employed to verify the correctness and feasibility of quaternion method in the calculation of beam propagation, and the pincushion distortion caused by the optical system was successfully simulated.

通过对某光学系统的仿真分析，成功仿真出图像经过光学系统产生的枕形畸变，证明四元数方法在光线传输运算中的正确性与可行性，为复杂光学系统的空间光线传输计算提供了另一种方法。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。