一类由“三角板旋转”演绎出的中考命题

所谓“旋转”就是在平面内,一个图形绕着某一点按一定的方向旋转一定的角度,这样的图形运动称为旋转,这一点叫做旋转中心,旋转是由旋转中心和旋转的角度决定的．由旋转的意义可知,旋转具有以下特征：(1)图形旋转时,图形上的每一点都绕着旋转中心旋转了同样大小的角度；(2)旋转后的图形与原来图形的对应线     

反比例函数比例系数的几何意义及应用

反比例函数比例系数的几何意义:反比例函数图像上的任意一点的横坐标与纵坐标的乘积都等于比例系数k的值,如图1所示．过双曲线上的一点A(m,n)作AB垂直于x轴,垂足为B,AC垂直于y轴,垂足     

为什么tan90°无意义?

<正>当我们被问到"为什么tan 90°无意义"时,总会下意识回答"因为正切函数的函数图象在90°处取不到",但事实上这是不准确的.对于三角函数而言,是先有三角函数的概念,之后才有的三角函数图象,因此并不能通过函数图象在某一点是否取得到,来解释函数在某一点是否有意义,而需要从函数的由来入手进行解释.三角函数源于天文学、测量等实际需要,正切函数和与之联系较为紧密的余切函数的研究主要是由于日影测量而引起的.     

高等数学教学中应重视几何直观的作用

在高等数学中几何方法与分析方法是密不可分的,分析法使问题严谨而富有哲理性,几何方法使问题形象直观,所讨论问题的几何意义对寻求问题的解具有启发性和指导意义.本文通过具体实例阐明了这一点.     

解题过程中字母意义的转化

许多数学问题中都会含有字母,但是字母在整个问题中所代表的意义并不是固定不变的,有时表示常量,有时表示变量;有时表示主变量,有时又表示参变量。对于中学生来说,认识到这一点需要有一个过程,引导学生观察和分析参数的意义是有益的。为此,下面我们通过例题对这     

Pre-Schwarz导数意义下区域之间的距离

Lehto曾用Schwarz导数定义了边界多于一点的两个单连通区域的Mbius等价类之间的"距离",并猜测它是一个距离.但最近Bozin和Markovic否定了这一猜想.一个自然的问题就是:在Pre-Schwarz导数意义相应情况如何?用Pre-Schwarz导数给出了边界多于一点的两个单连通区域的仿射等价类之间的"距离",并证明了这样定义的"距离"是一个伪距离,即使将其限制在由具有解析边界的单连通区域的仿射等价类空间上也是如此.     

用几何方法借助导数画一些曲线的切线

仅用初等数学只能画圆的切线,而导数概念纳入中学教材后,则可借助导数画出许多非圆曲线的切线,这对于学生理解导数的几何意义,了解导数的一个应用是有益的。本文将给出一些常用曲线上任意一点处切线的准确画法。 (一) 曲线y=e~x上一点P(x_0,y_0)处切线的画法。     

从一道隐函数求导的题说起

由方程yx=xy确定的函数y=y(x),用隐函数求导得到的y′(x)的表达式中,在x=e,y=e点无意义,是不是y(x)、y′(x)在这一点不存在?怎样理解“隐函数存在定理”?本文就这个问题进行探讨     

逻辑知识初步(续完)

§4 推理和论证 1.推理论证的意义和要求。推理是从一个或几个判断得出一个新判断的思维形式。例如: (1) 因为:三角形的内角和为180°;直角三角形是一种特殊的三角形。所以:直角三角形的内角和为180°。 (2) 因为:锐角三角形的三高交于一点;直角三角形的三高交于一点;钝角三角形三高的所在直线交于一点;三角形只有锐角三角形、直角三角形和钝角三角形三种可能情况。所以;三角形三高的所在直线交于一点。由此可见,推理由两个部分组成:一部分是推理所依据的判断,叫做前提:一部分是推出的新判断,叫做结论。论证是利用一些确实可靠的判断,通过推理,来阐明某个判断的真实性的过程。数学中的论证通常叫     

两个正三角形引出的几个结论

如图一,B为线段AC上一点,在AC的同向作正三角形ABD和正三角形BCE,这个构图虽然很简单,但却能得到多个有趣的结论.在以下的讨论中,同一个字母表示的意义相同,再次使用时不再重复说明.结论一如图一,连AE和DC,则AE=DC.     

Scalarization of Henig Proper Efficient Points in a Normed Space

In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base.     

Staircase Parameterization in Dynamical Selection

This paper deals with a variant of a dynamical selection scheme introduced by Attouch and Cominetti for ill-posed convex minimization which combines approximation with the steepest descent method by mean of a suitable parameterization of the approximation parameter as a function of the time. This variant applies to a general inclusion with a maximal monotone operator by mean of a staircase parameterization. A discrete analogue is also considered. Applications to selecting a particular zero of a maximal monotone operator or a particular fixed point of a nonexpansive mapping via regularization techniques are presented. Finally, the alternative use of well-posedness by perturbations is discussed.     

On arithmetic Macaulayfication of Noetherian rings

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

     

A Barrier Function Method for the Nonconvex Quadratic Programming Problem with Box Constraints

In this paper a barrier function method is proposed for approximating a solution of the nonconvex quadratic programming problem with box constraints. The method attempts to produce a solution of good quality by following a path as the barrier parameter decreases from a sufficiently large positive number. For a given value of the barrier parameter, the method searches for a minimum point of the barrier function in a descent direction, which has a desired property that the box constraints are always satisfied automatically if the step length is a number between zero and one. When all the diagonal entries of the objective function are negative, the method converges to at least a local minimum point of the problem if it yields a local minimum point of the barrier function for a sequence of decreasing values of the barrier parameter with zero limit. Numerical results show that the method always generates a global or near global minimum point as the barrier parameter decreases at a sufficiently slow pace.     

Genus fields of Kummer extensions of rational function fields

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.     

DC programming techniques for solving a class of nonlinear bilevel programs

We propose a method for finding a global solution of a class of nonlinear bilevel programs, in which the objective function in the first level is a DC function, and the second level consists of finding a Karush-Kuhn-Tucker point of a quadratic programming problem. This method is a combination of the local algorithm DCA in DC programming with a branch and bound scheme well known in discrete and global optimization. Computational results on a class of quadratic bilevel programs are reported.     

Counting dissections into integral squares

A squared rectangle is a rectangle dissected into squares. Similarly a rectangled rectangle is a rectangle dissected into rectangles. The classic paper ‘The dissection of rectangles into squares’ of Brooks, Smith, Stone and Tutte described a beautiful connection between squared rectangles and harmonic functions. In this paper we count dissections of a rectangle into a set of integral squares or a set of integral rectangles. Here, some squares and rectangles may have the same size. We introduce a method involving a recurrence relation of large sized matrices to enumerate squared and rectangled rectangles of a given sized rectangle and propose the asymptotic behavior of their growth rates.     

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing.In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.     

On a Mathematical Comparison between Hierarchy and Network with a Classification of Coordination Structures

This paper proposes a mathematical model to compare a network organization with a hierarchical organization. In order to formulate the model, we define a three-dimensional framework of the coordination structure of a network and of other typical coordination structures. In the framework, we can define a network structure by contrasting it with a hierarchy, in terms of the distribution of decision making, which is one of the main features of information processing. Based on this definition, we have developed a mathematical model for evaluating coordination structures. Using this model, we can derive two boundary conditions among the coordination structures with respect to the optimal coordination structure. The boundary conditions help us to understand why an organization changes its coordination structure from a hierarchy to a network and what factors cause this change. They enable us, for example, to find points of structural change where the optimal coordination structure shifts from a hierarchy to a hierarchy with delegation or from a hierarchy with delegation to a network, when the nature of the task changes from routine to non-routine. In conclusion, our framework and model may provide a basis for discussing the processes that occur when coordination structures change between a hierarchy and a network.