C~*-代数的Cuntz半群(英文)

令A是一个C~*-代数.设(x,y)是A的Cuntz半群W(A)中的一个元素对.本文在适当的条件下具体刻画了所有的(x,y),使其满足性质:如果x≤y,那么存在z∈W(A)使得x+z=y.另外,本文还讨论了交换C~*-代数关于Cuntz比较的一些性质.     

构造几何图形解竞赛题

解初中数学竞赛题的方法很多 ,有时使人觉得扑朔迷离 ,无从下手或解法太繁 .而构造几何图形解竞赛题却是十分巧妙的方法 ,也体现着数形结合的优越性 .构造图形解题的过程是一种创造性的思维过程 ,常伴随着观察、分析、综合、联想、猜想等思维活动 ,具有灵活性大 ,难度高、技巧性强等特点 .下面介绍构造几何图形来解竞赛题 .1.求极值( 1)已知x、y、z为正数 ,且 (x y) (y z)=2 ,试求xyz(x y z)的最大值 .分析 :由x、y、z为正数 ,又出现x y ,y z,故可构造边长为x y、y z、x z的三角形 ,由切线长定理可知 ,三角形内有一内切圆 .解 :如图 ,构…     

数学问题解答

1999年11月号数学问题解答(解答由问题提供人给出)1221.求方程组x y z=3x3 y3 z3=3的所有整数解.解　原方程组化为x y=3-z(1)x3 y3=3-z3(2)(1)3-(2),得3xy(x y)=24-27z 9z2(3)(1)代入(3),可得xy=8-9z 3z23-z(4)由(1)、(4)知x、y是以下二次方程的两个整数根:t2-(3-z)t 8-9z 3z23-z=0解得t1,2=3-z±(z-1)2·z 5z-32=3-z±(z-1)2(1 8z-3)2(5)由此知,x、y、z均为整数当且仅当z-1=0或z-3=1或z-3=-8,即z=1或z=4或z=-5.将其依次代入求根公式(5),得原方程组的所有整数解(共四组):x=1y=1z=1或x=-5y=4z=4或x=4y=-5z=4或x=4y=4z=-5注:(5)式中根号内的(z…     

关于Diophantine方程(20n)x+(21n)y=(29n)z

设a,b,c是满足条件a2+ b2=c2的两两互素的正整数.Jesmanowicz于1956年猜想对于任意给定的正整数n,方程(an)x+(bn)y=(cn)z仅有解(x,y,z)=(2,2,2).本文证明了方程(20n)x+(21n)y=(29n)z有唯一解(x,y,z)=(2,2,2).     

序Banach空间中非混合单调三元算子方程(组)解的存在唯一性

本文在序Banach空间中建立了再生正规锥条件下的非混合单调三元算子方程组{T_1(x,y,z)=x,T_2(x,y,z)=y,T_3(x,y,z)=z}以及三元算子方程T(x,x,x)=x解的存在唯一性定理,所得结果推广了已有文献中的二元算子方程(组)解的存在唯一性定理.     

指数Diophantine方程(bn)~x+(2n)~y=((b+2)n)~z的例外解

设b是大于3的正奇数.运用初等方法讨论了方程(bn)^x+(2n)x+(2n)^y=((b+2)n)y=((b+2)n)^z适合(x,y,z)≠(1,1,1)的正整数解(x,y,z,n).证明了:i)对于任何给定的正整数N,存在无穷多个b可使该方程有满足min{x,y,z}≥N的正整数解(x,y,z,n);ii)对于任何给定的b,该方程仅有有限多组正整数解(x,y,z,n)满足y>z=x.     

W·Janoux的一个不等式的简证

设 x、y、z∈ R ,证明 :y2 - x2z x z2 - y2x y x2 - z2y z ≥ 0 .此题就是著名的 W .Janoux猜想 ,最初发表在加拿大《数学难题》杂志 ,他本人没给出证明 .我国《中等数学》转载后引起关注 ,并给出了若干证法 ,但多数较烦 .下面给出一个简明 .证明　∵　 x、y、z∈ R ,∴　 y2 - x2z x z2 - y2x y x2 - z2y z　　 =y2 - x2z x (x - y) z2 - y2x y (y - z) x2 - z2y z (z - x)　　 =(y - x) (y - z)z x (z - y) (z - x)x y (x - z) (x - y)y z .但　 (y - x) (y - z)z x (z - y) (z- x)x y　 =(y- z) [(y- x) (y…     

三角形不等式的共同解法—关于三角形旁切圆半径的两个不等式证明的改进

众所周知(x y)(y z)(z x)=xy(x y) yz(y z) zx(z x) 2xyz=x2y xy2 y2z yz2 z2x zx2 2xyz　(*)这是一个十分重要的代数恒等式,由(*)立即得到(x y)(y z)(z x)=(x y z)(xy yz zx)-xyz(1)(x y)(y z)(z x)=x(y z)2 y(z x)2 z(x y)2-4xyz(2)(x y)(y z)(z x)(x y z)=xy(x y)2 yz(y z)2 zx(z x)2 4xyz(x y z)(3)(x y)(y z)(z x)(xy yz zx)=x2y2(x y) y2z2(y z) z2x2(z x) 2xyz(x y z)2(4)……(*)及(1),(2),(3),(4)……在证明关于三角形不等式方面有极其广泛的应用.这是因为:图1任一三角形总有内切圆(图1),总可以作变换a=y z,b=z x,c=x y(x,y,z∈R )…     

数学问题解答

20 0 4年 1月号问题解答(解答由问题提供人给出 )1 471　求方程组 x+y =ztz+t =xy的非负整数解 .解　因为方程组中x与y ,z与t可以互换 ,所以可以先求满足 0 ≤x≤y ,0 ≤z≤t的整数解组 (x,y ,z,t) .( 1 )若x、z中有一个为零 ,不妨设x=0 ,则由原方程组消去t得 :y+z2 =0所以y =z=0 ,t= 0 .即 ( 0 ,0 ,0 ,0 )是原方程组求的一组解 .( 2 )若x ,z都不是 0 ,但是有一个为 1 ,设x=1 ,则由原方程组消去y得 :t+z=zt - 1所以 (z- 1 ) (t- 1 ) =2 ,因为z,t为正整数且z≤t,所以z - 1 =1t- 1 =2 得z=2 ,t =3,y=5即 ( 1 ,5 ,2 ,3)是原方程组的一组解 ,同…     

在共振点附近的一类二阶泛函微分方程的解析解

在复域C内研究一类包含未知函数迭代的二阶微分方程x″(z)=G(z,x(z),x~2(z),…,x~m(z))解析解的存在性.通过Schr(?)der变换,即x(z)=y(αy~(-1)(z)),把这类方程转化为一种不含未知函数迭代的泛函微分方程α~2y″(αz)y″(z)-αy′(αz)y″(z)= (y′(z))~3G(y(z),y(αz),…,y(α~mz)),并给出它的局部可逆解析解.本文不仅讨论了双曲型情形0<|α|<1和共振的情形(α是一个单位根),而且还在Brjuno条件下讨论了共振点附近的情形(即单位根附近).     

Quasi-Minuscule Quotients and Reduced Words for Reflections

We study the reduced expressions for reflections in Coxeter groups, with particular emphasis on finite Weyl groups. For example, the number of reduced expressions for any reflection can be expressed as the sum of the squares of the number of reduced expressions for certain elements naturally associated to the reflection. In the case of the longest reflection in a Weyl group, we use a theorem of Dale Peterson to provide an explicit formula for the number of reduced expressions. We also show that the reduced expressions for any Weyl group reflection are in bijection with the linear extensions of a natural partial ordering of a subset of the positive roots or co-roots.     

PRODUCTS OF DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract

To form products of differential expressions in the classical way it is necessary to place heavy differentiability assumptions on the coefficients. Here we consider symmetric (formally self-adjoint) expressions defined, not in the classical way, but in terms of quasi-derivatives. With this very general notion of symmetry we show that products such as M1M2MI of symmetric expressions M1, Hp can be formed vithout any smoothness assumptions on the coefficients and such products are symmetric expressions.     

关于三元r-循环实矩阵及其应用

讨论三元 r-循环实矩阵 ,给出了三元 r-循环实矩阵的行列式和逆矩阵的实表达式 .从而得到r-循环实矩阵的行列式和逆矩阵的实表达式     

Algorithms from algebra

The rules of algebra are rules about classes of objects. In many cases, algebra can be viewed as algorithmic in nature. By specifically defining the class for which a rule applies and then the general outline of the rule, the guesswork can be removed from the process. Thus the student has a clearer understanding of when and how a particular rule applies. The paper describes this approach to the solution of algebraic expressions and the simplification of rational expressions.     

Modeling of coupling mechanisms and frequency separation in double disk resonators

We present modeling expressions for the coupling mechanisms in vertically stacked disk resonators. Using lumped modeling techniques, analytic expressions for coupling mechanisms in a stack of two disks are derived. The modeling results are compared to those of high-fidelity finite element analyses and show that the analytic expressions derived can be used to predict frequency separation between radial extensional modes. The established model is useful for gaining insight into parameter sensitivity when designing a stacked double disk resonator.     

THE NUMBER OF ROOTED NEARLY CUBIC C-NETS

1. IntroductionW.T. Tutte's original papers[1--3) on the enumerative theory of rooted planar maps havebrought forth a series of papers on enumerating triangulations. The enumeration of generalrooted planar maps has then also been investigated and a number of elegant results havebeen obtained, although relatively fewer than that of triangulations. As the dual case oftriangulations, the enumerative theory of cubic maps has also been developed, though thereare a lot of problems waiting for solut…     

Reducing trigonometric functions by using spokes on a carousel

In this note, a simple and quick method is shown for reducing some trigonometric expressions by using a diagram called a ‘carousel’. These expressions can be reduced using other methods, such as: (i) reduction formulae for sum/difference of angles; and (ii) graph shifting techniques.     

Calculation of stresses in a coal seam in presence of gas diffusion

Analytical expressions are obtained for calculating the stresses in a coal seam in presence of gas diffusion. The solution is given as a sum whose four terms are convergent series. All functions in the relevant expressions can easily be computed either separately or successively; i.e., it is not required to solve infinite systems of equations.     

Limit-Point Criteria For Not Necessarily Symmetric Ordinary Differential Expressions

A limit-point criterion for not necessarily symmetric differential expressions L of arbitrary order is proved by applying a known limit-point criterion for symmetric even-order differential expressions to (L⁺L)^p for some positive integer p where L⁺ denotes the formal adjoint of L. For certain differential expressions the results obtained from the new criterion are stronger than those obtained from the original one and can even be improved by increasing p.     

散度、旋度和梯度的统一定义

本文根据统一的共同模式导出了散度、旋度和梯度的微分表达式。所有这些量都是单位容积中某一微分量的极限值,在利用了单位矢量在度规系数加权下的某些微分共系式后,我们立刻就能导出矢量分析中这三种量的全部表达式。