形如“|x|+|y|=1”的画法及其扩展

人教版教材第二册(上)第八章第二节讲的是椭圆的简单几何性质,在对称性中有这样的描述,在曲线的方程中,如果以-y代y方程不变,那么当P(x,y)在曲线上时,它关于x轴的对称点P′(x,-y)也在曲线上,所以曲线关于x轴对称,同理如果以-x代x方程不变,那么曲线关于y轴对称,如果同时以-x代x,以     

躺着思想,不如站起行动

在很久很久以前,有两个年轻人,一起去遥远的地方寻找幸福和快乐,一路上风餐露宿,历尽艰辛,在即将到达目的地时,遇到了一条风急浪高的大河.对于如何渡过这条河,两人有不同的意见:一个人建议砍伐附近的树造一条木船渡过河去,另一个人则认为这么高的浪,这么湍急的水流,这么宽的河面,无论造多大的船都不可能渡过这条河,与其自寻烦恼和死路,不如等这条河干了,再轻轻松松地走过去.     

Low dimensional modules over quantum complete intersections in two variables

We classify all the indecomposable modules of dimension ≤ 5 over the quantum exterior algebra k(x, y)/(x^2, y^2, xy + qyx) in two variables, and all the indecomposable modules of dimension ≤3 over the quantum complete intersection k(x,y)/(x^m,y^n,xy + qyx) in two variables, where m or n ≥3, by giving explicitly their diagram presentations.     

论怀疑

现代世界,资讯发达,我们常常会听到一些匪夷所思的事情.这些传闻可信么?的确,大千世界,无奇不有,好像什么事情都可能发生,但我们也不能听到什么都相信我们要有怀疑精神,要独立思考,对该事件发生的可能性大小作一些分析,有助于我们作出理性判断.也就是说,我们要作出理性判断,要以怀疑精神作为前提,以概率分析为辅助.下面,就让我们从怀疑谈起.     

是谁拉你走向平庸

一名长跑运动员参加一个5人小组的比赛,赛前教练对他说,据我了解,其他4人的实力不如你,于是,这名运动员轻松地跑了第一名随后教练又让他参加了一个10人小组比赛,教练把平时其他人的成绩拿给他看,他发现别人的成绩并不如自己,他又轻松跑了第一名然后,这名运动员又参加了一个20人小组的比赛,教练说,你只要战胜其中的一个人,你就能取得胜利,结果,比赛中他紧跟着教练说的那个运动员,并在最后冲刺时,又取得了第一名.     

本期特约编委简介

杨仁宽,男,湖北恩施市人,湖北大学毕业.广东省数学特级教师,广东省南粤优秀教师,广东省名师工作室主持人,广州市名教师,教育部"国培研修员",省教育厅"南粤名师大讲堂"主讲、主评,广东省示范性高中广州市从化中学教研主任,从化市十大科技杰出人才,从化市劳动模范,《数学大世界》执行编委,国家级优秀(核心)期刊《高中数理化》特邀编委,广州市教育局第十三届特约教研员,主持教育部规划课题的子课题结题,主持广州市教育局资助课题在研,核心期刊《中学数学》2006年第10期     

抓住小条件 解决大问题

每一道习题都有着严密的逻辑性,已知条件不可能多余,也不可能短缺,在所有条件中,抓住其最有特征性的一个,联想展开,这是解题的一种途径．许多同学在解题时,往往不去认真推敲题目中给出的已知条件,对于一些细小的似乎是不起眼的说明,便不去深入探讨,弃之一旁,熟视无睹,这是一个极大的错误．很多时候,若能紧紧抓住这些小条件,便可从这里打开缺口,解决大问题．     

共同探究,享受数学

众所周知,数学很重要,这很大层面上是由高考指挥棒决定的,学生在沉重的数学作业负担下,大多是疲于应付,很少有人真正体验到数学内在的美,学会享受数学.笔者从事高中数学多年,做了一点有益的尝试,让学生从多种角度理解数学,师生平等,教师做好组织、引领工作,充分调动学生学习数学积极性,各抒己见,经常是一个平淡的问题,往往出现多种漂亮的解法,碰撞出思维的火花,师生乐在其中,一同享受数学带给我们的快乐.现举两个案例,供大家分享.     

品鸭脖赏小题——武汉市中考数学选择压轴题赏析

武汉鸭脖名扬天下,鸭脖虽小,品之有趣:一在麻劲够劲,二在慢啃细嚼之间回味无穷.同样,武汉人出的中考数学题也匠心独运,特色鲜明,出现在“脖颈”处的小题,带着股明显的鸭脖味儿,初品辣火呛口,但只要解题者细细品味、慢慢咀嚼,总会嚼出“香气”,唇齿留香.     

相似三角形等积式问题证明“三部曲”

等积式证明问题是初中平面几何的重要内容,它涉及的知识内容广泛,有利于培养学生综合运用知识的能力.等积问题综合性强,类型繁多,涉及面广,难度大,加之许多学生由于基础知识不牢,不善于归纳总结,结果在解决等积问题时不能灵活运用,感觉问题的分析困难,甚至是无从下手,望而生畏.为此,     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

Qualitative Analysis of Bifurcating Solutions in the Lengyel-Epstein Model

One of the most fundamental problems in theoretical biology is to explain the mechanisms by which patterns and forms are created in the＇living world. In his seminal paper ＂The Chemical Basis of Morphogenesis＂, Turing showed that a system of coupled reaction-diffusion equations can be used to describe patterns and forms in biological systems. However, the first experimental evidence to the Turing patterns was observed by De Kepper and her associates（1990） on the CIMA reaction in an open unstirred reactor, almost 40 years after Turing＇s prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. The Lengyel-Epstein model is in the form as follows     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

(0,1 ;0)-INTERPOLATION ON INFINITE INTERVAL (-∞, +∞)

In this paper, we study the explicit representation and convergence of (0, 1; 0)-interpolation on infinite interval, which means to determine a polynomial of degree ≤ 3n - 2 when the function values are prescribed at two set of points namely the zeros of Hn(x) and H′n(x) and the first derivatives at the zeros of H′n(x).     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].