例谈“以退求进”法解题策略

著名数学家华罗庚说过:善于退,足够的退,退到最原始而不失去重要的地方,是学好数学的一个诀窍.这里所谓的退,当然不是逃跑,而是养精蓄锐,蓄势待发,是在为进寻求途径,即以退为进.它的实质是借助转化的数学思想,把复杂的问题简单化,运动的问题静止化,高维问题低维化,变量问题常量化,抽象问题具体化,代数问     

空集的自述

我姓空,我的名字叫空集,我是不含任何元素的集合,可谓腹中空空,一无所有,不过你们别小觑我啊,我的本事大着呢!不信,我说出来,别吓你一跳哟!1.特殊的我我是集合大家庭中的一员,不含任何元素的     

真假四边形市长风波

在一个名叫几何的王国里,有一座繁荣的小城—凸多边形城,这里住着各种各样的多边形.城中有一个四边形家族,成员非常多,如不规则的四边形一家,平行四边形一家,梯形一家等等.其中平行四边形一家地位特殊,因为这个凸多边形城市的市长便是那大名鼎鼎、无人不晓的平行四边形.平行四边形处事公平,执政已有多年,在它的带领下,凸多边形城市才发展成为一片欣欣向荣的景象.平日里,平行四边形市长最喜     

聚焦高考平面向量题

一、近几年平面向量考题的特点特点一:考小题,重在基础.有关平面向量的小题,其考查的重点在于基础知识:其中,平面向量数量积、加减运算是考查的重点,有关向量共线,向量垂直,向量的模,坐标运算等内容的试题都突出了对平面向量基础知识的考查.特点二:考大题,与其他知识结合.有关平面向量的大题,经常与三角、圆锥曲线、函数等结合,与三角函数相结合的试题难度不大,属中档题,与圆锥曲线、函数相结合的试题,属中等偏难,主要考查学生对基本知识,基本方法,基本技能的理解,掌握和应用情况.     

有得有失 有加有减

正林泽宇说,他的爸爸妈妈经常吵架,他很羡(xiàn)慕(mù)其他同学有温暖和谐的家庭,其实,很多家庭都有不如意的事情,就拿我家来说,我爸爸是做生意的,他经常在全国各地飞来飞去,很少有时间陪我,为此我经常闷闷不乐。昨天早上,爸爸匆匆忙忙地整理好行李就去机场了,谁知过了半天,他又回来了,原来,由于雾太大,航班取消了。     

依靠自己站起来

以前,在西方的一个国家里,有一个经理,他把所有积蓄全部投资在了一个项目上,但由于世界大战的爆发,他破产了.他认为是他把家人害得没有了一切,于是就离开家人,成了一名流浪汉.他心情非常沮丧,对往事难以释怀,不肯为以后的日子打算,甚至想要自杀.一天,他偶然看到了一本名为《自信心》的书,这本书的内容讲的是人在生活、工作上崩溃了以后,如何重新恢复信心的事.他看后心里又有了一丝希望,想找到作者,请作者帮他再度站起来.     

例说数学思维的困境及其摆脱

思维受挫就是在解题时,按一般的思路介入后,或者浅尝辄止,或者眼花缭乱,或者误入歧途,一错再错,或者前途渺茫,无法推进,就算使尽了浑身解数,左冲右突,终觉“山重水复疑无路”.笔者以为,要修复受挫的思维,只要仔细读题审题,寻找蛛丝马迹,类比联想相关知识点,分析可能的思想方法,探索恰当的思维路径,就会“峰回路转,柳暗花明”.下面举例说明,供同学们参考.     

《高效课堂理论与实践》学术研讨会邀请函

来吧,请到这里来,这里有知音,这里有琴台;来吧,请到这里来,这里有黄鹤,这里有辛亥。来会会你的数学知音,来寻找你的心中至爱;来听听黄鹤的古老传说,来弘扬首义的精神气概。来推动教育改革的新潮,来领略高效课堂的风采;来观摩精英们的科研成果,来见证获奖者的青春光彩;     

换个想法,奇迹也许就会出现

有一个美国人,他的妻子是医生,在非洲做援助工作,他准备去看她.妻子在信中告诉他,那里除了一些当地土著人,就是荒芜的土地,没有交流,没有娱乐,该有的这里都没有,一起来的人大部分都受不了而中途落跑了,让他做好充分的准备.他不相信.但到了后才发现,当地的生活环境比想象的还要糟糕.他和妻子生活在荒漠中的小屋里,不会当地土著语言,离开翻译,寸步难行.而翻译也只是在有病人时,才陪着病人出现.这里无人对话,没有事做.走出小屋,就是光秃秃的土地.晚上到     

走出学案导学的误区,打造和谐高效的课堂

近年来,由于学案的方便好用,兼有教案的作用,被许多教师采用.笔者所在的学校适应“减负增效”的新形式,大力推进学案导学,打造高效课堂.但是,在实施学案导学的过程中,由于部分教师对“学案导学”理解不到位,在实施过程中,没有发挥学案应有的效能,走入了一些误区.笔者以一个案例为例,结合自己在实施学案导学过程中的一些体会,谈谈如何走出学案导学的误区,打造和谐高效的课堂.     

Convergence of densities of some functionals of Gaussian processes

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein?s method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein–Uhlenbeck processes is discussed.     

L－fuzzy集网的R－收敛性质

本文在ＬＦ拓扑空间中建立了Ｌ－ｆｕｚｚｙ集网的弱收敛（Ｒ－收敛）概念，应用文［４］中的Ｒ－闭包，系统讨论了它们的性质，证明了等式ＲｌｉｍＡ＿ｎ＝∧（∨Ａ＿ｍ）＿Ｒ和ＲｌｉｍＡ＿ｎ＝Ａ＿ｎ＝∧（∨Ａ＿ｍ）＿Ｒ并且给出了Ｌ－ｆｕｚｚｙ集网与其子网之间的关系。     

Colength of varieties of linear algebras

In this paper, an example of a variety of nonassociative algebras is constructed in which the system of Capelli identities of small rank is satisfied, whereas the colength function has exponential growth. As is well known, in the associative case and in the Lie case, the validity of Capelli identities implies the polynomial growth of the colength.     

The evolution of the concept of homeomorphism

Topology, or analysis situs, has often been regarded as the study of those properties of point sets (in Euclidean space or in abstract spaces) that are invariant under “homeomorphisms.” Besides the modern concept of homeomorphism, at least three other concepts were used in this context during the late 19th and early 20th centuries, and regarded (by various mathematicians) as characterizing topology: deformations, diffeomorphisms, and continuous bijections. Poincaré, in particular, characterized analysis situs in terms of deformations in 1892 but in terms of diffeomorphisms in 1895. Eventually Kuratowski showed in 1921 that in the plane there can be a continuous bijection of P onto Q, and of Q onto P, without P and Q being homeomorphic.     

Examples of the influence of the geometry on the propagation of progressive waves

In this paper, we give examples of the influence of the domain of propagation on progressive waves. More precisely, we numerically investigate the propagation of reaction diffusion waves in cylinders with variable radius. We show that, when the radius rapidly expands from a very small radius to a larger one, depending on the viscosity and the nonlinearity, the travelling wave may be blocked. The aim of this paper is to give numerical illustrations and quantifications of this effect, and to propose some conjectures which could be interesting subjects for further mathematical investigations.This work is linked to the study of spreading depression (SD), a propagative mechanism in brain and various tissues which has been observed in vivo and in vitro in many species since their discovery in 1944 by Leao. As a matter of fact, their direct observation in Man is still controversial. The complex structure of gray and white matter in humans may block the propagation of SD over large distances in brain and thus explain the difficulty of observing it. Medical consequences of the current numerical studies are detailed in [M.A. Dronne, et al., Influence of brain geometry on spreading depressions: A computationnal study, Progress in Biophysics and Molecular Biology 97 (1) (2008) 54–59] and a first mathematical approach given in [M.A. Dronne, E. Grenier, H. Gilquin, Modelization of spreading depressions following Nedergaard, preprint, 2003].     

Unitarizability of a certain class of irreducible representations of classical groups

We prove that a certain class of irreducible representations of the classical p-adic groups is unitarizable and in general, can be isolated in the unitary dual. These representations are Aubert duals of a certain class of square-integrable representations, thus, in this case, Bernstein’s conjecture, which states that the Aubert involution preserves unitarizability, is confirmed.     

Analysis of the Stress State of the Surface Layer of a Polymeric Mass upon Filling of Bulky Compression Molds

The stress state of the surface layer of a polymeric mass during filling of bulky compression molds is analyzed. It is shown that, at particular rheological characteristics of the mass, temperature, and filling rates, cracking of the surface layer occurs, which leads to defects in the finished products. A physical analysis of this process makes it possible to conclude that the cracks arise due to the normal stresses operating in the front region of the moving polymeric mass. It is found that, under certain flow conditions, areas with a pressure lower than the atmospheric one appear on the surface of the polymer. If the tensile stresses arising in these local regions are higher than the tensile strength of the mass, the continuity of the composition is broken in the direction determined by the greatest rate of the normal deformation. To confirm the reliability of the crack-formation mechanism proposed, the distribution of the pressure and normal stresses over the free surface is calculated based on a numerical method. These calculations show that, by comparing the stress level achieved in the front region with the tensile-strength characteristics of the polymeric composition, it is possible to predict, with a sufficient accuracy, the possibility of crack formation in the surface layer of such a mass under given flow conditions and thus to solve the question on flawless manufacturing of products.     

Origin of the division of labour and a stochastic mechanism of differentiation

Nonlinear models are introduced to describe the nonequilibrium dynamics of social evolution. The difference between Western and Oriental culture, and their roles in the origin in the division of labour, are described by a behavioural model in information diffusion and learning competition. It shows a tradeoff between stability and diversity. The stochastic mechanism of social differentiation and the empirical evidence for this is discussed in a stochastic model of multi-staged development. It shows the breakdown of the Gaussian distribution during a transition. Finally, an ideal model of social bifurcation is given.     

模糊层次分析法在矩阵论教材评价方面的应用

基础学科的教材直接影响学生的基本功,尤其是几乎每个学科都会涉及的数学类教材,如矩阵论.矩阵论是研究生的基础课程,在对学生以后的学术道路有举足轻重的作用.所以选择一本合适的教材,对学生和教师来说都有不小的帮助.然而,对教材的评价而言,不能单单从一个方面入手,因此将模糊综合分析法与层次分析法结合,在矩阵论教材的评价方面建立评价体系,为高校选择合适的教材提供依据.     

Robust counterparts of inequalities containing sums of maxima of linear functions

This paper addresses the robust counterparts of optimization problems containing sums of maxima of linear functions. These problems include many practical problems, e.g. problems with sums of absolute values, and arise when taking the robust counterpart of a linear inequality that is affine in the decision variables, affine in a parameter with box uncertainty, and affine in a parameter with general uncertainty.