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.     

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.     

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。     

《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;     

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.     

纪念李国平老师百年诞辰——支持新兴学科 提倡科研创新

李国平老师是中国科学院首批学部委员(后称院士)。他一生勤于学习,刻苦钻研,对半纯函数论的发展做出了卓越的贡献,写出了许多极具价值的论文和著作,由国内外著名的期刊和出版社登载出版,深受学术界同行的瞩目和关注。李老师精心培育后辈,桃李满天下,50年代他的学生中有四位中科院院士、以及分布在各条战线的众多杰出人材,为国家建设做出各自的贡献,深受人民爱戴。他眼光远大,站     

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.     

有资格限制的指派问题的求解方法

在实际的指派工作中，常会遇到某个人有没有资格去承担某项工作的问题，因此，本建立了有资格限制的指派问题的数学模型。在此数学模型中，将效益矩阵转化为判定矩阵，由此给出了判定此种指派问题是否有解的方法；在有解的情况下，进一步将效益矩阵转化为求解矩阵，从而将有资格限制的指派问题化为传统的指派问题来求解。最后给出了一个数值例子来说明这样的处理方法是有效的。     

Admissibility of estimators of the probability of unobserved outcomes

The problem of estimating the probability of unobserved outcomes or, as it is sometimes called, the conditional probability of a new species, is studied. Good's estimator, which is essentially the same as Robbins' estimator, namely the number of singleton species observed divided by the sample size, is studied from a decision theory point of view. The results obtained are as follows: (1) When the total number of different species is assumed bounded by some known number, Good's and Robbins' estimators are inadmissible for squared error loss. (2) If the number of different species can be infinite, Good's and Robbins' estimators are admissible for squared error loss. (3) Whereas Robbins' estimator is a UMVUE for theunconditional probability of a new species obtained in one extra sample point, Robbins' estimator is not a uniformly minimum mean squared error unbiased estimator of the conditional probability of a new species. This answers a question raised by Robbins. (4) It is shown that for Robbins' model and squared error loss, there are admissible Bayes estimators which do not depend only on a minimal sufficient statistic. A discussion of interpretations and significance of the results is offered. Research supported by NSF Grant DMS-88-22622.     

22号初等元胞自动机的极限语言复杂性

§1Introduction Thetaskofthispaperistosolvetheproblemproposedin[1],i.e.,toexplorethe complexityoflimitset,orequivalently,limitlanguageoftheelementarycellularautomaton ofrule22bythetoolsofformallanguagetheory.Cellularautomata(abbreviatedasCA),asmathematicalmodelsforcomplexnatural systemscontaininglargenumbersofsimpleidenticalcomponentswithlocalinteractions,havebeenwidelyusedinphysical,biological,chemicalandcomputationalsystems[2].Despitetheirsimpleconstruction,someCAcandisplayveryrichandcompl…     

Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.     

The method of determining all real nonmultiple roots of systems of nonlinear equations

A method for determining all nonmultiple roots of the system of nonlinear equations in an n-dimensional parallelepiped is proposed. The main idea of the method is that the original set, in which the roots are sought, is divided into subsets where either the system of equations does not have solutions or its Jacobian matrix is nonsingular. A partition algorithm is presented and its convergence is proved. The application of the method is demonstrated using several examples.     

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs $G$ and $H$. Let $H^k\left[G\right]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k\left[G\right]$ and $H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k\left[G\right]$ and $H^k$ for $G$ and $H$ arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10¹⁰⁰ ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.     

有限子集系的Sperner系

1９８０年，Ｋｏ-Ｗｅｉ Ｌｉｈ提出如下猜想：如果Ｆ是由Ｂ^ｎ中固定秩的不同元素生成的序理想，那么Ｆ是Ｓｐｅｒｎｅｒ系．本文证实了当Ｆ是由Ｘ的子集Ｙ的所有相同秩的元素生成的序理想，猜想是正确的     

On estimating of the number of constituents of a finite mixture of continuous distributions

Summary Suppose thatH is a mixture of distributions for a given familyF A necessary and sufficient condition is obtained under whichH is, in fact, a finite mixture. An estimator of the number of distributions constituting the mixture is proposed assuming that the mixture is finite and its asymptotic properties are investigated.     

万有Teichmuller空间各分支边界点的一些几何性质

在对数导数意义下,万有Teichmuller空间T1可表示为无穷多个互不相交的连通分支的并集.本文研究了该模型各分支的几何性质,给出了为e-iθ/（1-e-iθz）为L和Le的公共边界点,且在‖·‖1的意义下,证明了L,L0,Lθ两两公共边界点之间的距离均为2.