尽力讲清重要概念产生的背景——关于二次曲线不变量教学的点滴体会

关于一般二次曲线方程的讨论,是解析几何中理论性较强的内容之一,它是中学数学二次曲线标准方程的推广,又是进一步学习一般二次曲面方程的讨论的基础和准备。一般二次曲线方程的讨论,包括用坐标变换的方法化简一般二元二次方程为标准形式,从而对一般二次曲线进行分类,并作出图形。还包括用不变量来判别一般二次曲线的类型,以及用不变量直接写出化简后的方程,从而确定曲线的形状和大小。在这两种方法中,坐标变换的方法是基本的,不变量     

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.     

二次曲线方程的一种化简方法

在解析几何里利用坐标轴的旋转和平移来化简二次曲线方程是一种基本的方法.下面我们来介绍一种化简二次曲线方程的方法,这种方法十分简便.只根据四条简单的引理就能解决问题,我们先介绍这四条引理.     

确定抛物线位置的简捷方法

所见书刊都是先用坐标变换等理论,将一般二次曲线方程化简为标准方程,从而确定曲线形状和位置的,如文[1]是用“基本不变量”化简方程确定曲线的位置;而文[2]是用二次曲线的直径方程研     

谈无中心型二次曲线方程的化简

数学通讯一九八九年第九期发表了《也谈二次曲线的化简问题》,该文通过圆锥曲线的直径方程给出了中心型曲线 u(x,y)=Ax~2+Bxy+Cy~2+F=0化简的一种方法。本文将利用二次曲线的对称     

二次曲线方程的化简与坐标变换

中学教材的二次曲线部分增添了坐标轴旋转的内容。坐标轴的平移与旋转的主要作用之一,就是为了将二次曲线的一般方程化为最简方程。现就有关内容作些补充与说明,便于学习这一段教材时参考,弄懂在化简过程时如何作和为什么要这样作。一、有关二次曲线的三个命题     

二次曲线的一种化简方法

二次曲线化简通常采用转轴劝移轴的方法。本刊1980年第4期[1]介绍了抛物型二次曲线的一种化简方法,但对椭圆型、双曲型二次曲线并未论及。本文拟给出二次曲线化简的另一种简捷方法,其实质为待     

二次曲线和蚶线——一个有趣的联系

在直角坐标系中讨论二次曲线的图象及其性质是平面解析几何的重要内容之一,同样,在极坐标系中讨论蜡线也是平面解析几何中不可缺少的一部分.表面上看二次曲线和蜡线是没有联系的,然而在蜡线的倒数方程里     

利用特征根研究二次曲线的切线问题

根据二次曲线依其特征根所作的分类,分别讨论各类二次曲线的特征根与其切线方程一般式各系数之间的具体关系,从而获得任意一条直线是否为二次曲线切线的充要条件。     

直线方程f(2x0-x,2y0-y)=f(x,y)的几何意义

文 ( 1 )给出了直线方程 x0 x y0 y =r2的几何意义 ,文 ( 2 )又给出了直线方程 x0 xa2 y0 yb2 =1的几何意义 ,两文的讨论仅涉及到圆和椭圆这两种最简单的标准方程 ,本文将把这种讨论推广到一般的常态二次曲线 .设常态二次曲线 L的方程为 f( x,y) =0 ,M( x0 ,y0 )为坐标平面内任一点 ,本文讨论下列方程 ( * )的几何意义 .f ( 2 x0 - x,2 y0 - y) - f( x,y) =0　 ( * )定理 1　设 M( x0 ,y0 )为常态二次曲线L :f ( x,y) =0内部一点 ,那么方程 ( * )的几何意义表示以点 M为中点的中点弦所在的直线 .证明　在曲线 L :f ( x,y) =0上任取一…     

Computing approximate solutions for convex conic systems of constraints

We propose a way to reformulate a conic system of constraints as an optimization problem. When an appropriate interior-point method (ipm) is applied to the reformulation, the ipm iterates yield backward-approximate solutions, that is, solutions for nearby conic systems. In addition, once the number of ipm iterations passes a certain threshold, the ipm iterates yield forward-approximate solutions, that is, points close to an exact solution of the original conic system. The threshold is proportional to the reciprocal of distance to ill-posedness of the original conic system.?The condition numbers of the linear equations encountered when applying an ipm influence the computational cost at each iteration. We show that for the reformulation, the condition numbers of the linear equations are uniformly bounded both when computing reasonably-accurate backward-approximate solutions to arbitrary conic systems and when computing forward-approximate solutions to well-conditioned conic systems. Received: July 11, 1997 / Accepted: August 18, 1999?Published online March 15, 2000     

A Third Derivative Estimate for Monge-Ampere Equations with Conic Singularities

The author applies the arguments in his PKU Master degree thesis in 1988 to derive a third derivative estimate,and consequently,a C2,α-estimate,for complex MongeAmpere equations in the conic case.This C2,α-estimate was used by Jeffres-MazzeoRubinstein in their proof of the existence of K（a）hler-Einstein metrics with conic singularities.     

A conic bundle description of Moishezon twistor spaces without effective divisors of degree one

In [K2] Moishezon twistor spaces over the connected sum (), which do not contain effective divisors of degree one, were constructed as deformations of the twistor spaces introduced in [LeB]. We study their structure for by constructing a modification which is a conic bundle over . We show that they are rational. In case n = 4 we give explicit equations for such conic bundles and use them to construct explicit birational maps between these conic bundles and . Received October 8, 1996; in final form May 16, 1997     

On the copositive representation of binary and continuous nonconvex quadratic programs

In this paper, we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices. This result can be viewed as an extension of earlier separate results, which have established the copositive representation of a small collection of NP-hard problems. A simplification, which reduces the dimension of the linear conic program, and an extension to complementarity constraints are established, and computational issues are discussed. Research partially supported by NSF Grant CCF-0545514.     

Structure of the Integrals of Equations of Vibration for a Conic Shell Closed at the Vertex

Ukrainian Mathematical Journal - We consider a system of differential equations that describes free oscillations of a thin-walled conic shell of revolution with vertex. On the basis of the analytic...     

On the solvability of one multidimensional version of the first Darboux problem for some nonlinear wave equations

In the present paper, for wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of a multidimensional version of the first Darboux problem in the conic domain.     

Equations of <Emphasis Type="Italic">G</Emphasis>-minimal conic bundles with degree 4

We consider the first nontrivial case of relatively G-minimal conic bundles which are G-minimal and have r = 4 singular fibers. Classification obtained gives explicit equations of minimal conic bundles (S, G) and an explicit action of the group G on the Picard group Pic(S) and on the surface S.     

Return-mapping algorithms for associative isotropic hardening plasticity using conic optimization

We present a mathematical programming approach for elastoplastic constitutive initial value problems. Consideration of the associative plasticity and a linear isotropic hardening model allowed us to formulate the local discrete constitutive equations as conic programs. Specifically, we demonstrate that implicit return-mapping schemes for well-known yield criteria, such as the Rankine, von Mises, Tresca, Drucker-Prager, and Mohr–Coulomb criteria, can be expressed as second-order and semidefinite conic programs. Additionally, we propose a novel scheme for the numerical evaluation of the consistent elastoplastic tangent operator based on a first-order parameter derivative of the optimal solutions.     

On the solvability of one boundary value problem for some semilinear wave equations with source terms

In a conic domain of time type for one class of semilinear wave equations with source terms we consider a Sobolev problem representing a multidimensional version of the Darboux second problem. The questions on global and local solvability, uniqueness and absence of solutions of this problem are investigated.     

Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds

We prove the global-in-time Strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds. We obtain estimates for the full set of wave admissible indices, including the endpoint. The key points are the properties of the microlocalized spectral measure of Laplacian on this setting showed in [18] and a Littlewood–Paley squarefunction estimate. As applications, we prove the global existence and scattering for a family of nonlinear wave equations on this setting.