一道高中数学预赛题的多向推广

众所周知,等差数列存在一些美妙的性质,列出如下. 性质1 等差数列{an｝的前n项之和An=an2+bn. 性质2 若等差数列{an｝与等差数列{bn｝前n项之和分别为An,Bn,则An/Bn=an+b/cn+d. 证明:设{an｝,{bn｝的公差分别为d1,d2,由An=na1+n(n-1)d1,Bn=nb1+n(n-1)d2,得An/Bn=d1/2n+(a1-d1/2)/d2/2n+(b1-d2/2)=an+b/cn+d,其中a=d1/2,b=a1-d1/2,c=d2/2,d=b1-d2/2.     

关于圆外切闭折线的几个不等式

设△ABC的边长为a ,b,c,其内切圆为⊙ (I,r)即“圆心为I,半径为r” ,则有下面的不等式成立 (证略 ) :AI2 +BI2 +CI2 ≥ 14(a2 +b2 +c2 ) + 3r2( 1 )文献 [1 ]中还有以下不等式 :AI+BI+CI≥ 6r ( 2 )( 1 )、( 2 )中等号成立当且仅当a=b=c.本文将推广这两个不等式 ,证明关于圆外切闭折线的几个不等式 .定理 1　设平面闭折线A1 A2 A3…AnA1 的内切圆为⊙ (I,r) ,其边长为 ｜AiAi+1 ｜=ai(i=1 ,2 ,… ,n ,且An+1 为A1 ) ,则有∑ni=1AiI2 ≥ 14∑ni=1a2 i+nr2 ( 3)当且仅当a1 =a2 =… =an 时取等号 .证明　设已知闭折线的边AiAi- 1 …     

柯西不等式的证明及应用

<正>高中数学学习中,不等式变形巧妙神奇,尤其是柯西不等式的应用.我梳理了一下有关柯西不等式的证明及应用,方便同学们使用.柯西不等式:(a1b1+a2b2+…+an bn)2≤(a21+a22+…+a2n)(b21+b22+…+b2n)(ai bi∈R,i=1,2…n).等号当且仅当a1=a2=…=an=0或bi=tai时成立(t为常数,i=1,2…n).柯西不等式的证明方法很多,下面的方法比较深刻且具通性.为简便,设ai不全为0.证法一(构造二次函数)f(x)=(a1x+b1)2+(a2x+b2)2+…+(an x+bn)2=(a21+a22+…+a2n)x2+2(a1b1+a2b2+…+an bn)x+(b21+b22+…+b2n).     

考点39 数列的极限

1.(浙江卷,1)limn→∞1+2+3+…+nn2=().(A)2(B)1(C)21(D)02.(湖南卷,3)已知数列{log2(an-1)}(n∈N*)为等差数列,且a1=3,a2=5,则limn→∞(a2-1a1+a3-1a2+…+an+11-an)=().(A)2(B)23(C)1(D)213.(广东卷,10)已知数列{xn}满足x2=x21,xn=21(xn-1+xn-2),n=3,4,….若limn→∞xn=2,则x1=().(A)23(B)3(C)4(D)54.(山东卷,13)limn→∞Cn2+2Cnn-2(n+1)2=.5.(重庆卷,14)limn→∞23n-32n+123n+32n=.6.(上海卷,7)计算limn→∞3n+1-2n3n+2n+1=.7.(福建卷,15)若常数b满足b>1,则limn→∞1+b+b2+…+bn-1bn=.8.(天津卷,18)已知un=an+an-1b+an-2b2+…+abn-1…     

初二代数第一学期期末自测题

A组一、选择题 (每小题 3分 ,共 3 0分 )1 .下列由左边到右边的变形 ,属于因式分解的是(　　 ) .A .( 2a +b) ( 2a -b) =4a2 -b2B .x2 -2x -3 =(x +1 ) (x-3 )C .4x2 y3 -2xy2 +1 =2xy2 ( 2xy -1 ) +1D .a2 +1 =a(a+1a)2 .若 4x2 +2 (k-4)x+1是一个完全平方式 ,则k的值为 (　　 ) .A .4和 2　　　　B .5和 2C .6和 2　　　　D . -4和 23 .下列各式中 ,变形不正确的是A .c-ab=-cab　　　B .-b-3a=b3aC .3b-4a=-3b4a　　D .--7b3a =7b-3a4.若 4a =5b(b≠ 0 ) ,则 a2 -b2b2 的值等于 (　　 ) .A .-15 　　　B .14　　　C .91 6　　　D .…     

一个积分的计算

在这篇文章里我们将计算下面积分.此积分在研制移动通讯过程中被用到[1].我们将用留数算出它的完整形式.In=∫∞0…∫∞0∫π20exp-x1+2x2+…+nxn2sin2yexp-a1x1-(a1+a2)x2-…-(a1+a2+…+an)xndx1…dxndy.令b1=a1,b2=a1+a2,…,bn=a1+a2+…+an,可得In=∫π20∫∞0…∫∞0exp-x112sin2y+b1exp-x222sin2y+b2…exp-xnn2sin2y+bndx1dx2…dxndy=∫π20∫∞0exp-x112sin2y+b1dx1∫∞0exp-x222sin2y+b2dx2…∫∞0exp-xnn2sin2y+bndxndy=∫π20Θ1Θ2…Θndy,其中Θk=Θ(k,bk)=∫∞0exp-xkk2sin2y+bkdxk.今有Θ(a,b)=∫∞0exp-xa2sin2y+bdx=2sin2ya+…     

一个n有序集组的计数问题

In this paper, we discuss the counting problem of an order n-group of set (A1 ,A2 ,… ,An) which satisfies ∪^ni=1Ai={a,a2,…,am} and one of the following:(1)∩^ni=1Ai=Φ;(2)∩^ni=1Ai={b1,b2,…,bk};(3)∩^ni=1A1包含{b1,b2,…,bk};(4)Ai≠Φ(i=1,2,…,k），We solve these problems by element analytical method.     

一道数学竞赛题的推广

设 a1,a2 ,… ,an和 b1,b2 ,… ,bn都是非负实数 ,则　 [( a1+ b1) ( a2 + b2 )… ( an + bn) ]1n ≥　 ( a1a2 … an) 1n + ( b1b2 … bn) 1n.这是第 6 4届普特兰数学竞赛中的一道题目 .本文给出该不等式的一个推广 .推广　设 aij >0 ( i =1 ,2 ,… ,m;j=1 ,2 ,… ,n) ,则( ∑mi=1ai1∑mi=1ai2 …∑mi=1ain) 1n ≥ ∑mi=1( ai1ai2 … ain) 1n,等号当且仅当 as1at1=as2at2=… =asnatn( s,t=1 ,2 ,… ,m;s≠ t)时成立 .证明　由平均不等式知 :1na11∑mi=1ai1+ a12∑mi=1ai2+… + a1n∑mi=1ain≥　a11a12 … a1n∑mi=1ai1∑mi=1ai2 …∑mi=1ai…     

一个不等式的拓广

文 [1],[2 ]均对不等式“已知 :a >0 ,b >0 ,a3 +b3 =2 ,则a +b≤ 2”作出了一系列的讨论 .本文将给出该不等式的两个拓广 ,并由此证明了文 [2 ]末给出的猜想命题 1　若an +bn=2 ,a ,b∈R ,n≥ 2且n∈N ,则a +b≤ 2 ,ab≤ 1.上述命题为原不等式在指数上的推广 ,即文 [2 ]中猜想 1.证　 1)当a >0 ,b >0时 ,∵an+bn≥ 2anbn ,∴ 2anbn ≤ 2 ,即anbn≤ 1.∴ab≤ 1.又an+1+… +1n -1个 1+bn+1+… +1n -1个 1≥n nan +n·nbn,即na +nb≤ 2 +2 (n - 1) ,∴a +b≤ 2 .2 )若a <0 ,b <0 ,由题设n必为偶数 .此时 ,an+bn=(-a) n+(-b) n=2 .由 1)知 :(-…     

一道东南数学奥林匹克试题的推广

题目(第三届(2006年)东南数学奥林匹克第6题):求最小的实数m,使不等式m(a3 b3 c3)≥6(a2 b2 c2) 1(1)对满足a b c=1的任意正实数a,b,c恒成立.本文给出此题的一个推广.推广设ai>0,i=1,2,…,n,n≥2,∑ni=1ai=1,B>0,A>-Bn,求最小的实数m,使不等式m∑ni=1ai3≥A∑ni=1ai2 B(2)恒成立.注:在推广中取n=3,A=6,B=1即得上述东南竞赛题.解ai=1n,i=1,2,…,n,得m≥An Bn2.下面证明,当ai>0,i=1,2,…,n,n≥2,∑ni=1ai=1,B>0,A>-Bn时,有(An Bn2)∑ni=1ai3≥A∑ni=1ai2 B(3)下面证明(3)式成立.不妨设a1≥a2≥…≥an,则a12≥a22≥…≥an2,由切比雪夫不…     

Schur algebras of Brauer algebras I

Schur algebras of Brauer algebras are defined as endomorphism algebras of certain direct sums of ‘permutation modules’ over Brauer algebras. Explicit combinatorial bases of these new Schur algebras are given; in particular, these Schur algebras are defined integrally. The new Schur algebras are related to the Brauer algebra by Schur–Weyl dualities on the above sums of permutation modules. Moreover, they are shown to be quasi-hereditary. Over fields of characteristic different from two or three, the new Schur algebras are quasi-hereditary 1-covers of Brauer algebras, and hence the unique ‘canonical’ Schur algebras of Brauer algebras.     

可积代数正规类中半素代数类及半素一致代数类确定的上根

利用代数正规类中的理想乘积公理，引入可积代数正规类及可积代数正规类中素代数、半素代数类及一致代数类概念，讨论了可积代数正规类中半素代数类及半素一致代数类确定的上根性质。     

Centroids of Zinbiel Algebras

Zinbiel algebras were introduced by a way of Koszul dual to Leibniz algebras and connected with commutative associate algebras, pre-Lie algebras and dendriform algebras etc. In this article, the centroids of Zinbiel algebra are discussed. We study the proprieties of centroids and also determinate the centroids of Zinbiel algebras up to dimension ≤4. As consequent, we get the conclusion that the centroids of 2-dimensional Zinbiel algebras are not small and 3,4-dimensional Zinbiel algebras are almost small.     

Representations of Frobenius-type triangular matrix algebras

The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.     

One-relator quotients of free products of cyclic groups

Abstract

The question of existence of n-ary systems playing the role of enveloping algebras for Filippov (n-Lie) algebras is investigated. We consider the ternary algebras whose reduced binary algebras are associative, the associative algebras, and two other classes of ternary algebras as possible enveloping algebras.     

一类特征单的n-李代数

利用结合代数与n-李代数的张量积构造了一类无限维特征单的n-李代数,且证明了除n=3的情形以外,这类特征单n-李代数的内导子代数是特征单李代数.     

Foundations of the Theory of Dual Lie Algebras

In this paper we introduce and study dual Lie algebras, i.e., Lie algebras over the algebra of dual numbers. We establish some fundamental properties of such Lie algebras and compare them with the corresponding properties of real and complex Lie algebras. We discuss the question of classification of dual Lie algebras of small dimension and consider the connection of dual Lie algebras with approximate Lie algebras.     

Enveloping Algebras of Solvable Malcev Algebras of Dimension Five

We study the universal enveloping algebras of the one-parameter family of solvable 5-dimensional non-Lie Malcev algebras. We explicitly determine the universal nonassociative enveloping algebras (in the sense of Pérez-Izquierdo and Shestakov) and the centers of the universal enveloping algebras. We also determine the universal alternative enveloping algebras.     

一类非Hopf 代数的双Frobenius 代数

本文构造了一类非Hopf 代数的双Frobenius 代数. 特别地, 在某些特殊的情形下, 这里构造的双Frobenius 代数是整体维数为3 的阶1 生成的Artin-Schelter 正则代数的Yoneda 代数.     

Inductions and restrictions on towers of cellularly stratified diagram algebras

Hartmann et al. defined the concept of cellularly stratified algebras that combine the features of both cellular algebras and stratified algebras. Many important diagram algebras in mathematics and physics, such as some Brauer, partition and BMW algebras, are cellularly stratified algebras, and each of these forms a tower of algebras. This article gives the concept of towers of cellularly stratified algebras in an axiomatic manner, and studies it in terms of induction and restriction functors. In particular, for certain towers of cellularly stratified algebras, we provide a criterion for semi-simplicity by using the cohomology groups of cell modules.