四边形面积的一个性质及其应用

定理凸四边形的两条对角线把四边形划分成的四个小三角形中,两组对顶的两个三角形面积之积相等。证明如图1,记∠AOB=a,△AOB、△COD、△AOD和△BOC的面积分别为S_1,S_2,S_3和S_4,则由三角形面积公式,可知 sin(180°-a)。故得S_1S_2=S_3S_4。在图1中,若AB∥CD,则S_△ACD=S_△BCD,可见S_3=S_4,再据定理,有     

一道高中联赛题的简解

1993年度全国高中数学竞赛有一这样一个题目:实数x、y满足4x~2-5xy 4y~2=5,设s=x~2 y~2,则1/S_(max) 1/S_(min)的值为______.本文给出它的两个仅用到初中知识的解法。     

关于Segal代数的乘子

设G为局部紧交换群,为G的对偶群.设S_1(G)与S_2(G)是G上的Segal代数.记S_1(G)到S_2(G)的乘子全体为M(S_1,S_2).本文主要证明了下面两个结果: 1.T∈M(S,L~1)当且仅当存在唯一的σ∈E_s~*使得Tf=σ*f f∈S(G),且‖T‖=‖σ‖E_s~*. 2.设S_2(G)S_1(G)且‖f‖S_1≤‖f‖S_2,f∈S_2(G).若T∈M(S_1,S_2),则存在唯一的G上有界连续函数φ使得其中是f的Fourier变换.     

灵机一动

<正>贵刊2017年8月下期18页刊登了王建荣和刘沙西两个老师的文章《剖析一道几何题》,通过作平行线,得出了结论,其实,这样的解法看起来巧妙,实绕一大弯.这里给出一个更直接的证明方法.题目P是平行四边形ABCD中的任意一点.求证:S_(△ABP)=S_(△APD)+S_(△ACP).从图上可以看出:S_(△ABP)+S_(△CPD)=(1/2)S_(平行四边形ABCD);S_(△APD)+S_(△ACP)+S_(△CPD)=(1/2)S_(平行四边形ABCD).     

四边形中的趣味竞赛题(下)

<正>例9在任意给定的凸四边形ABCD中,边AB,BC,CD,DA的中点分别为E,F,G和H.求证:四边形ABCD的面积≤EG×HF≤1/2(AB+CD)×1/2(AD+BC).证明如图11所示,HE∥DB∥GF,又EF∥HG,所以EFGH为平行四边形.S_(ABCD)=S_(EFGH)+S_(△AEH)+S_(△DGH)+S_(△CGF)+S_(△BEF),而S_(△AEH)+S_(△CGF)=1/4(S_(△ABD0)+S_(△CBD))=1/4S_(ABCD).同理可证S_(△DGH)+S_(△BEF)=1/4S_(ABCD),所以S_(ABCD)=S_(EFGH)+1/2S_(ABCD),     

分担集合的亚纯函数正规族

对复平面C的非空有限子集S_1和S_2,记在复平面区域D内满足{z∈D:f(z)∈S_1}={z∈D:f′(z)∈S_2}的全体亚纯函数f形成的函数族为D,那么当S_1和S_2共有至少12个元素对函数族D正规.特别地,当S_1具有至少三个复数时,我们得到了准确的结果.     

CM分担两个公共值集的亚纯函数唯一性

证明了存在一个具有2个元素的集合S_1和一个具有5个元素的集合S_2,使得对任何两个非常数亚纯函数f与g,只要满足E(S_j,f)=E(S_j,g)(j=1,2),就有f≡g.     

培养学生探究能力的一次尝试

笔者曾为培养学生的探究能力作了一次尝试,收到良好的效果。现介绍于下。 (一)在竞赛中激发兴趣为了激发学生的兴趣,我选取了个人竞赛的方法: 1、自己取数a、b、c组成集合A={-1、a、b、c}; 2、求A中所有数之和S_1; 3、在A中任取两数作积,它们的和记作S_2; 4、在A中任取三数作积,它们的和记作S_3; 5、求A中四个数之积S_4; 6、计算S_1 S_2 S_3 S_4的值。要求:计算结果要保密,算出来的同学要举手。我把算出来的同学名字按先后次序写在黑板上(对学生是鼓励),等大部分同学举手后,再叫他们说出计算的结果-1。     

联图S_5∨C_n的交叉数

两个图G和H的联图,记作G∨H,是指将G中每个点与H中的每个点连边得到的图.本文证明了星图S_5与圈C_n的联图S_5∨C_n的交叉数为Z(6,n)+4[n/2]+3(n≥3),其中Z(m,n)=[m/2][(m-1)/2][n/2][(n-1)/2],m,n为非负整数.     

可分的共形循环黎曼空间的分类

本文得到的结果是可分的共形循环空间 C_n 共分四类:(Ⅰ)C_n=R_n 是 Car-tan 意义下的循环空间;(Ⅱ)C_n=S_n(?)×S_(?)是共形平坦空间;(Ⅲ)C_n=V_2×S_(π-z)是非常曲率2维空间的常曲率扩充;(Ⅳ)C_4=V′_2×V″_2是两个非常曲率2维空间的乘积空间.     

A family of nonisomorphic Markov random fields

It was recently shown that there exists a family of ℤ² Markov random fields which areK but are not isomorphic to Bernoulli shifts [4]. In this paper we show that most distinct members of this family are not isomorphic. This implies that there is a two parameter family of ℤ² Markov random fields of the same entropy, no two of which are isomorphic.     

与线性变换的完全环同构的环理论(Ⅰ)

<正> 熟知地,满足极小条件的单纯环只与一个有限维向量空间的线性变换的完全环同构.并且此向量空间如取为左向量空间的话,那末R的任一极小右理想均可取为此左向量空间.在没有有限条件情况下,Jacobsoo用本原环来取代这种单纯环.接着Wolfson研     

The Average Eccentricity of Sierpiński Graphs

We determine the eccentricity of an arbitrary vertex, the average eccentricity and its standard deviation for all Sierpiński graphs ${S_p^n}$ . Special cases are the graphs ${S_2^{n}}$ , which are isomorphic to the state graphs of the Chinese Rings puzzle with n rings and the graphs ${S_3^{n}}$ isomorphic to the Hanoi graphs ${H_3^{n}}$ representing the Tower of Hanoi puzzle with 3 pegs and n discs.     

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

Noether’s problem for GL(2, 3)

We prove that Noether’s problem has an affirmative answer for the group GL(2, 3), over every field K. In particular, the group admits a generic polynomial over . As a consequence, so does the group . Research was partially supported by MCYT grant BFM2003-01898.     

Existence of EF‐equivalent non‐isomorphic models

We prove the existence of pairs of models of the same cardinality λ which are very equivalent according to EF games, but not isomorphic. We continue the paper [4], but we do not rely on it. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximal nests in the Calkin algebra

We prove that if two countable commutative lattices of projections in the Calkin algebra are order isomorphic, then they are unitarily equivalent. We show that there are isomorphic maximal nests of projections in the Calkin algebra that are order isomorphic but not similar. Assuming the continuum hypothesis, we show that all maximal nests of projections in the Calkin algebra are order isomorphic.

     

A formalization of Sambins's normalization for GL

Sambin [6] proved the normalization theorem (Hauptsatz) for GL, the modal logic of provability, in a sequent calculus version called by him GLS. His proof does not take into account the concept of reduction, commonly used in normalization proofs. Bellini [1], on the other hand, gave a normalization proof for GL using reductions. Indeed, Sambin's proof is a decision procedure which builds cut-free proofs. In this work we formalize this procedure as a recursive function and prove its recursiveness in an arithmetically formalizable way, concluding that the normalization of GL can be formalized in PA. MSC: 03F05, 03B35, 03B45.     

最高阶元素个数为40的非可解群的分类

设$\varphi$为群${\rm Aut}(N)$的同态,记$H_\varphi\times N$为群$N$借助于群$H$的半直积.设$G$为有限不可解群,本文证明: 若$G$中最高阶元素个数为40, 则$G$同构于下列群之一:(1)~$Z_{4\varphi}\times A_5$,\,${\rm ker}\varphi=Z_2$; (2)~$D_{8\varphi}\times A_5,\,{\rm ker}\varphi=Z_2\times Z_2$; (3)~$G/N=S_5$, $N=Z(G)=Z_2$; (4)~$G/N=S_5$, $N=Z_2\times Z_2,\,N\cap Z(G)=Z_2$.