n元椭球面的判定及所围椭球体的体积

给出n元二次曲面是椭球面的充要条件和所对应的n维椭球体的体积计算公式,并且条件和体积的计算中只用到曲面中的系数行列式,使判定和体积计算较为方便.     

二次曲面图形的一种新作法

根据二次曲面方程的特征根与其图形主轴之间的关系可化简二次曲面的一般方程为标准方程。从而作出曲面的图形。     

二次曲面的构造式方程

本文根据二次曲面的中心点坐标、主轴方向、半轴长这几个几何参数直接构造曲面方程,并称之为二次曲面的构造式方程．     

四面体的十二点二次曲面

垂心四面体中四条高的垂足,四个面的重心及从各顶点与四面体的垂心连线的三等分点,共十二个点共球.试图把垂心改为四面体内的任意点,相应地把四条高线改换为过该点与每个顶点连线的共点直线组时,则将把垂心四面体的十二点球有趣地推广为四面体的十二点二次曲面.     

椭球体与抛物锥体的体积推导

在研究性学习中 ,受教材《立体几何》中用祖日恒原理推导球体的体积方法的启发 ,本文探索用祖日恒原理推导椭球体和抛物锥体的体积 .1　椭球体的体积椭球体即指将椭圆绕其一条对称轴旋转一周 (半周亦可 )所得的几何体 .下面构造恰当的几何体使用祖日恒原理推导其体积 .先研究半椭球的体积 .设椭球体是由长半轴长为ａ ,短半轴长为ｂ的椭圆绕其短轴所在直线旋转得到的 .为了应用祖日恒原理 ,需要找一个可求体积的几何体 ,使它和旋转半椭球体可夹在两平行平面间 ,用平行于这两个平面的任意一个平面去截它们时 ,截面面积总相等 .图 1　求半椭球…     

二次曲面的几个性质

介绍了二次曲面的切线的几个性质 ,把二次曲线的类似性质推广到了二次曲面 .     

二次曲面理论的一个应用

本文使用二次曲面的化简理论 ,对于给定的三根螺旋 ,求出由它们确定的单叶双曲面     

平面与二次曲面相切的充要条件

针对平面与二次曲面相切的条件。根据二次曲面标准方程的五种分类．利用二次曲面上点的切平面的唯一性．给出任一平面与五类二次曲面相切的充要条件，得到一组相关推论，并给出一些应用实例．     

二次曲面的截面面积公式

一般二次曲面被任一平面所截得的截痕一定是椭圆．通过构造Lagrange函数求条件极值的方法。可得有心二次柱面被过原点平面所截得的椭圆面积公式．据此并结合空间解析几何理论,将一般二次曲面的平面截痕视作其射影柱面与该平面的截痕,进而可得一般二次曲面被任一平面截得的平面域的面积公式．     

二次曲面直纹性的工程应用

高等数学教材中关于二次曲面形状的讨论都采用截痕法.教学中引入直纹面概念,可为建筑施工专业的学生密切联系工程实际提供理论依据,收到较好的教学效果.     

用正交变换化实二次型为标准形的同步求解问题

作为《关于矩阵的特征值与特征向量同步求解问题》的续篇,利用其给出的方法,证明了新的定理.通过对实对称矩阵进行行列互逆变换,同步求出二次型的标准形及正交变换阵,简化了复杂的施密特正交化法,较好地解决了二次型标准形与正交变换阵同步求解问题.     

n维二次曲面的切超平面

利用特征根研究n维二次曲面的切超平面问题,给出平面为n维二次曲面的切超平面的充要条件.     

二次曲线方程的分类与化简

通过对二次曲线方程配方变形,利用直线与二次曲线相交时参数t的几何意义,以及仿射变换的性质,得到了二次曲线方程分类与化简的一种新方法,从而解决了二次曲线方程通过坐标系的平移、旋转进行分类、化简运算复杂,通过不变量进行化简,无法画出图形的具体位置等问题.     

Frobenius Groups Generated by Quadratic Elements

An automorphism of a group X is said to be quadratic if there exist integers and such that for any . If is a Frobenius group then an element is said to be quadratic if induces, by conjugation in the core of , a quadratic automorphism. By definition, a group H acts on a group F freely if for and only with or . It is proved that a Frobenius group generated by two quadratic elements is finite and its core is commutative. In particular, any Frobenius group generated by two elements of order at most 4 is finite. Also we argue that a Frobenius group with finitely generated soluble core is finite. The results mentioned are used to show that a group acting freely on an Abelian group is finite if it is generated by elements of order 3, and the order of a product of every two elements of order 3 in is finite.     

A Test-Example of a Quadratic Lattice

We compute the subdirectly irreducible factors of the free Hermitean (semi)lattice generated by a single element a subject to the relation aa ; we also compute the factors of its induced distributive presentation.     

基于Mathematica的旋转曲面教学实践研究

本文针对在旋转曲面教学实践中学生较难理解的一些空间旋转曲面问题,利用数学软件Mathematica分析了空间曲线绕任一直线旋转的具体过程,以及求解旋转曲面方程的方法,给出了求旋转曲面的具体步骤,流程图,以及Mathematica代码,并结合具体实例分析旋转过程.     

Finding Symmetry Groups of Some Quadratic Programming Problems

Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help decrease the problem dimension, reduce the size of the search space by means of linear cuts. While the previous studies of symmetries in the mathematical programming usually dealt with permutations of coordinates of the solutions space, the present paper considers a larger group of invertible linear transformations. We study a special case of the quadratic programming problem, where the objective function and constraints are given by quadratic forms. We formulate conditions, which allow us to transform the original problem to a new system of coordinates, such that the symmetries may be sought only among orthogonal transformations. In particular, these conditions are satisfied if the sum of all matrices of quadratic forms, involved in the constraints, is a positive definite matrix. We describe the structure and some useful properties of the group of symmetries of the problem. Besides that, the methods of detection of such symmetries are outlined for different special cases as well as for the general case.     

Quadratic decomposition of the symmetric semi-classical polynomial sequences of odd class: some examples from the class three

In this present work, we deal with the quadratic decomposition of symmetric semi-classical polynomial sequences of odd class. Some examples from class three are settled. The recurrence coefficients, the moments and the integral representations of the corresponding regular forms are established.